Sendo from Slam Dunk is great at basketball but is always late for meetings and hardly ever looks serious.
Axis Powers Hetalia: Ancient Rome is said to have conquered most of Europe, but is shown being almost more spacey than his own grandson, North Italy. He even spaces out when giving Germany a lecture on his own history.
There's also America. Childish, insane, and knows virtually nothing about the other nations he has to work with, yet he has built himself up to be very powerful and influential.
Tabane Shinonono of Infinite Stratos created the titular Powered Armor at the age of 15, but she acts like an hyperactive idiot in public, and LITERALLY wears bunny ears and dresses up as Alice from Alice in Wonderland. Her first public appearance in the anime has her visiting the resort her sister is at by riding a carrot-shaped Drop Pod down to the resort from who knows how high. Not to mention her unique methods of performance testing the IS her little sister is using for the first time: first simple acceleration, then testing out the included swords, then summoning a huge SAM battery from hammerspace which promptly fires a RobotechingMacross Missile Massacre of real missiles with live warheads at Houki.
"Next, try intercepting these!" (cue missile barrage) "There you go!"
Would you trust the former most notorious gang leader of your country, who just happens to be a pervert, a Man Child who is also One of the Kids, a slacker, and a klutz whose M.O. is destruction everywhere? That's exactly what Holy Forest Academy did when they hired Eikichi Onizuka to be a high school teacher in Great Teacher Onizuka. It turns out his unorthodox method of teaching is quite possibly the greatest thing to happen to the school.
Happosai of Ranma ˝ fame. He's constantly seen skirt-chasing, peeking, panty-stealing etc. but he is still one of the, if not the most powerful martial artist around. Same goes for his pupils Soun and especially Genma, who trained Ranma, who in turn has Bunny-Ears Lawyer streaks himself.
Belldandy of Ah! My Goddess is extremely competent and powerful with goddess magic, but woefully ignorant of most everything else, to the chagrin of her Love Interest, Keiichi.
Baccano!'s Claire Stanfield, an assassin that is very good at tracking people down, extracting information, and killing things. The catch is that he has a huge list of quirks (constant name-changing, a complete disregard for contract confidentiality, a habit of going wayoverboard, a firmly sustained belief that he is God — the list goes on) that can make him more than a little annoying for his employers to deal with. Still, he is really, really good at what he does.
Misato Katsuragi, from Neon Genesis Evangelion, is an unabashed drunk who lived in squalor until Shinji Ikari moved in with her (after which Shinji cleaned up after her) and stocks her fridge and pantry with tons of beer, instant noodles, and instant curry. Nonetheless, she is a military genius whose unconventional plans served well in fighting unconventional enemies such as the Angels. Said plans include, but do not limit to, defeating one Angel using a BFG powered by Japan's entire power grid, and defeating another Angel by having two pilots fight it in a synchronized dance.
Nizuma Eiji, mangaka prodigy of Bakuman。, can't work without making lots of loud sound effects and preferably turning up the stereo to battlefield volume. His way of blowing just about everybody off makes Topher look like a master of tact in the workplace. Altogether, the T.O. team seems to have branched out by making the Bunny Ears thoroughly obnoxious.
Tower of God: Phonsekal Laure is a narcoleptic, lazy bum who sleeps almost all the time, takes pillow and blanket anywhere and is one of the most talented Shinsoo users of the entire cast. Learning and deepening that skill is in universe the most time-consuming and difficult tasks for a Regular and everybody sure appreciates his skills.
Dr. Black Jack. The global medical community isn't quite sure how to feel about the greatest surgeon alive dressing like a vampire, chewing out half of his clients, and obstinately refusing to get a license because it limits the ridiculously high fees he charges. Nonetheless, most hospitals find themselves paying hundreds of thousands of dollars to sneak him in for an operation at least once. Also, Black Jack has, on two separate and unrelated occasions, been mistaken for the leader of an alien invasion.
Urahara back when he was still the Captain of Squad 12. The only shinigami scientist better than Mayuri and just as eccentric. To date, he's the played the role of the Eccentric Mentor, the Gadgeteer Genius, and will use Obfuscating Insanity as a weapon or just for fun. Even after he stopped being a captain, he was known as a strange business owner. He's so competent the first two-thirds of the manga can be viewed as one long chess game between him and Aizen. Doesn't stop him being an utter Cloud Cuckoo Lander when he feels like it, such as telling Ichigo to survive his training by putting on a silly bandana and gloves and yelling out a summons for the "hands of justice"... just because he felt like it.
The 11th division's reputation is simple: They're rough, they're violent, they're uncouth, they're undisciplined, they're sweaty, filthy, loincloth wearing men (to quote Isane and Rangiku). And then there's Yumichika. He fusses about his appearance, he worries about his looks. He will delay entering battle if he needs to change his clothes first. And he will kill you without a second thought if you call him ugly. He gets away with it because he's officially the third strongest man in the entire division and is capable of defeating vice-captains in battle (and Word of God has stated that he's Ikkaku's equal in strength, which unofficially makes him the joint second strongest man in the entire division).
Lieutenant Yachiru Kusajishi is the only known female in the 11th division, a small Rose-Haired Sweetie, Genki Girl. There's no way she looks like she should be there... and she spends most of her time doing almost anything under the sun for sweets. Then you notice that every single person in the entire division except the captain is absolutely terrified of her. As are the communication corps. (With damn good reason.)
Mashiro's childish, ditzy, naive and prone to temper tantrums. It's hard to believe she could ever become, let alone remain, a lieutenant of the Gotei 13 and yet, until the hollowfication accident, that is exactly what she was.
Captain Harlock has the Arcadia's entire crew. Oddly, Captain Harlock and Kei Yuki are the only ones who remotely seem to be serious about the whole pirating thing. Harlock later explains it: the Arcadia is an home before being a pirate warship, so the pirates acts just like they were in their previous homes. The two rules of the Arcadia, in fact are "You are in the crew until Harlock says so" and "If there's no emergency, do whatever you want". But when there is an emergency, the usually drunk, sleeping and lazy pirates becomes sober, serious and extremely competent fighters the very second they hear the alarm siren.
Lloyd Asplund is a goofy, graceless social misfit who is bluntly honest and speaks of everyone (even the Emperor) in an almost mocking tone. He's also one of the most brilliant engineers in the world, being the creator of the Super Prototype Lancelot. It probably doesn't hurt that he's an Earl.
L, the most brilliant detective in the world, is a reclusive and eccentric young man who locks himself away in an apartment and solves cases from around the world. He has awkward mannerisms and almost no social skills. He's always crouching in his chair and staring fixedly at something. He grips everything from the top with two fingers, like you might hold up a salamander or wet noodle. He's a Big Eater with an incredible sweet-tooth. He's almost always toying with or devouring a mountain of candy or pastries.
Mello and Near come from the same school for gifted children as L and are basically proto-Ls. Like L, they also have their eccentricities. With their powers combined, they are as good or better than L, and appropriately they split his habits: Near is always toying with things, in this case actual children's toys. Mello is always eating candy, in this case chocolate.
Muten Roshi of Dragon Ball. A 300-year oldperverted man... who happened to be the creator of the Kamehameha, tutor of Goku and Krillin and in early parts of the series the greatest martial artist on Earth.
In Dragon Ball Z, the Ginyu Force are a team of extremely campy, silly, overconfident apparent idiots who have odd quirks like striking embarrassing poses before a fight, betting chocolate bars on a fight's outcome, and playing Rock-Paper-Scissors to decide who gets to fight. However, there is a reason Frieza hired them as enforcers and ordered them to take down Vegeta. They are just that good at what they do.
Mr. Takuma of Elf Princess Rane is a brilliant Jack-of-All-Trades who insists on speaking backwards. All the time. If you look closely at the subtitles, he says at one point, "If I talk backwards, people will think I'm smart!"
Nanvel, from Burn Up!: she can make some incredible things, and has won several awards for her inventions, but, they're always designed in some absurd way that's supposed to be 'cute'. She also has a massive sweet-tooth, eats a certain snack exactly at 3:00 PM, never misses dessert, and loves children's programming.
Eyeshield 21 has a host of eccentric but skilled football players:
Yoichi Hiruma has a penchant for firearms of all kinds, and never hesitates to discharge them in the general direction of his teammates. He also enjoys heaping verbal and physical abuse upon them, to the point of kicking people he likes. And let's not get started on his use of blackmail to help his team. However, he is a highly skilled quarterback and tactician.
Otawara, also of the White Knights, is extremely slow-witted and crude; Running gags involve him farting and his pants falling down. Oh, and, while he isn't the most powerful lineman around, he's close, and he's incredibly fast for someone his size.
Jo Tetsuma of the Seibu Wild Gunmen almost never speaks unless Kid tells him to, and follows orders to the very syllable, even to illogical extents. In addition, he is only shown in two situations to take actions under his own initiative. However, his single-minded obedience to commands makes him one of the best four wide receivers in all of Kanto.
Kengo Mizumachi of the Kyoshin Poseidons is a spastic, energetic, irrepressibly cheerful guy who is constantly making a fool of himself but couldn't care less. Also, the only reason he started playing football is because no other sport provided an adequate level of challenge for him.
Agon Kongo is a dark take on the trope, along with being a kind of insufferable genius. He never trains and spends his days lazing about, picking fights, womanizing, and giving his brother (the team captain, no less) noogies, yet he gets away with it because he's the most talented athlete of the century.
The five hand-selected members of Mustang's personal unit are all excellent soldiers (for example, Breda graduated from the military academy at the top of his class), most of whom just act goofy. Mustang himself may also qualify, especially given his Obfuscating Perversity.
Armstrong also counts. He tends to be a tad... wacky, but it's not long before you realize it's a good idea not to piss him off.
Mithril appears to employ a number of these. There's Kurz Weber, a skilled AS pilot, top-grade sniper, and a complete Lovable Sex Maniac... and Teletha Testarossa, a sixteen-year-old dojikko with a bit of inferiority complex who's also a Teen Genius, the captain and designer of the Tuatha de Danaan, and the others' commanding officer. Sousuke, meanwhile, is a consummate professional when he's in a military setting... and a hopeless Fish out of Water with No Social Skills in any non-military situation.
In the novels, Sousuke himself takes on elements of the trope even within the military setting. He uses part of Mithril's military maneuvering ground to house his giant pet tiger (whom he referred to as a cat in his report to Tessa, making her largely regret approving it), and along with Kurz ends up completely trashing multi-million dollar Arm Slaves on a pirate treasure hunt. These and other shenanigans - including the incident from Ending Day By Day which did make it into the anime at the end of The Second Raid, in which Sousuke outright demands that Mithril's high command reinstate him as Kaname's bodyguard - are tolerated because Sousuke is the only person who can operate the Arbalest and its Lambda Driver, which are the only weapons Mithril has that can equal Amalgam's technology.
Graham Aker is pretty much what happens when an Otaku gets into military... except this Otaku is just about as Badass as he can get, to the point that he gets special permission to act as he wants, because his superiors figured out bad things tend to happen to their enemies when Graham gets to do things his way.
Andrew Waltfeld, the Desert Tiger, in Gundam SEED. He takes his coffee making very, very seriously, to the point of boiling it in an elaborate chemistry-set-looking contraption, talking about percentages of ingredients he's changed, and randomly mentioning it before or during serious combat situations. However, he's also a brilliant and decorated military commander and an ace mobile suit pilot, so his soldiers don't seem to do more than just blink in temporary confusion when he offers them the chance to try his latest brew.
RauLeCreuset of the same show manages to pretend to be one of these. He's The Obi-Wan to the deuteragonist, Athrun, and a senior military commander with the ear of the high-command...who happens to wear a white mask on all occasions and speak in cryptic, philosophical terms. No one seems to mind as his perceived competence, strategic ability and sheer Badass outweigh these considerations. And then he decides to drop theact...
Char Aznable himself is a bit of an oddball for constantly wearing that mask on his face, never removing it in front of anyone, including superior officers, even for a second. On the other hand, he's among the top aces and better commanders Zeon has, so nobody bugs him about it. The mask becomes such an iconic part of his persona that he's allowed to wear it to keep up the legend.
The mask is Justified in Mobile Suit Gundam The Origin by him having oversensible eyes and the mask both containing protective lenses and being harder to lose in action than normal glasses (in fact it had been originally given him at the military academy for that very reason by a fellow cadet who played rugby and took inspiration by similar masks used in that sport). Everything else, however, makes him straddle the line between Crazy Awesome (when Jaburo base was built, he worked on it. That's how he knew how to infiltrate it) and this trope (the infamous red paintjob).
Martian Successor Nadesico: Captain Yurika is a ditz, but also far more talented at her job than given credit for. Engineer Uribatake is a skirt chaser, and he signed on to get away from his wife, but he's also capable of designing absurdly-powerful machines. The pilots include a yaoi fangirl and a woman who loves puns—and her ukelele. Oh, and their lawyer has been known to wear bunny ears.
Legend of Galactic Heroes: Prototype and master to the above, Yang Wen-Li, a social misfit with an alcohol problem who almost flunked out of military academy where he only entered because he had no money to pay for his education to become a historian. He sits on his flagship's instrument panel during important fights (when he is not simply sleeping) and claims to everyone that he hates working. No one, no one ever wonders how he became admiral by 30; his skills make it obvious to any character (ally or enemy) that he deserves to be at the top of the chain of command.
Mihara Ichirou from Kidou Tenshi Angelic Layer acts every inch the Mad Scientist (even wearing a lab coat everywhere he goes), torments his assistant, never gives out his plans to his co-workers, and pops up around a twelve-year-old girl and drags her around town to play a doll fighting game; however, he's also a genius in robotics and, in the anime, an innovative medical researcher. That doll fighting game he drags the girl to? He created it. And not to mention computers that look like cute, teenage girls.
Kakashi Hatake is chronically late, reads porn (sometimes even in mid-battle), and is a bit of a goofball, but threaten his students or anyone else under his care and he'll kick your backside all over Fire Country.
Mr. Kishimoto seems to like these: Such as Gai and Lee who are Love Freaks and have Big Ol' Eyebrows but they're very strong and noble. Tsunade, who likes to drink and gamble, hates paperwork to a degree most people would reserve for their work enemy, and is a bit of a Vain Sorceress but also a competent leader as well as a kickass medic and fighter; Jiraiya, who's a self-proclaimed "super pervert" and porn writer but one of the strongest ninja in the world (to the point that some of the other top-tier ninja are wary of fighting him even with a numerical advantage) and who dies heroically. Anko (Stripperiffic and smartass kunoichi who was among the survivors of Orochimaru's experiments with Cursed Seals) and Shikamaru (genius strategist who is so lazy he'd rather hide his smarts than do paperwork) might also count.
Killer Bee tops all the others, though. He fights using seven swords, none of which he holds in his hands, because he prefers to keep them free so that he can write rap lyrics in the middle of combat.
Kisimoto seems to follow a certain trend with his characters: the stronger they are the more strange quirks they have, or in the worst cases (like Madara), the more Ax-Crazy they are, but they are dang good at what they do.
In Nerima Daikon Brothers there is a character named Ichiro who is the head host of a successful host club...who grows algae in a jar as a hobby and is in love with a panda.
The Straw Hat crew are a team of Bunny-Eared Lawyers. Here's the list, in the order in which they joined the crew: Zoro, first mate, is obsessed with strength, training, and swordsmanship cannot find south with the help of a bird whose head always points south, and can cut steel with a sword (and that was before the timeskip); Nami, the navigator, is unbelievably competent at her job and guessing weather changes, can use well a weather-based weapon to call thunder upon any enemy, and is one of the two relatively sane members of the crew, but is also incredibly greedy; Usopp is a liar, a pessimist and a coward, but is a skilled artists, his lies have proven themselves useful many times, when necessary he has done acts of insane bravery, he's an hell of a Combat Pragmatist who can take on far stronger enemies, his pessimism made him immune to the powers of an enemy who could turn anyone else in a depressed wreck, and he's the goddamn' best sniper in the world; Sanji is a womanizer who only fight with kicks to protect his hands, but his hands are what he uses as a Supreme Chef and his kicks put him on the Monster Trio of the crew with Luffy and Zoro; Chopper, the doctor, is a shape-shiftingtalking reindeer who is incredibly shy, and is both an incredibly skilled doctor and a fighter formidable enough to fight with the strongest opponents in the first half of the Grand Line and win; Nico Robin, the historian, is a Nightmare Fuel Station Attendant, a former assassin and brutally honest, but is also the sanest and most mature member of the crew and possibly the best historian in the world, and a formidable combatant; Franky, the carpenter, is a perverted speedo-wearing cyborg with a penchant for crying out "SUPER!", and has built himself into a cyborg and built the Cool Ship of the crew; Brook, the musician, is a a living skeleton with terrible puns and an obsession for women's underwear, who is also a genious musician, a formidable swordsman and used to be the captain of his own crew.
Some of the Marine higher ups, like Aokiji, who sleeps standing up and rides a bicycle across stretches of the ocean... which he froze with his powers, or Monkey D. Garp, Luffy's grandfather and the man who cornered the Pirate King Gold Roger several times throughout his career, and who baffles his men with his disregard for protocol and the rules. All of the senior staff at Impel Down are prime candidates too.
The Water 7 arc has an abundance of this trope. First there are the Galley-La shipwrights, who are the best in the world at what they do but count amongst their best a man with a rectangle nose who speaks like he's 80, another who never speaks at all outside of using a pigeon as a ventriloquist dummy and a secretary who attacks anyone who speaks ill of the boss (and ends up beating him up as well). Then those three reveal those were cover identities and they are actually government assassins (although still perfectly competent shipwrights) and rejoin the rest of their cell, who are all just as quirky. Of special note is Jabra, who is probably the weirdest of them all, but the only one who thinks assassins should dispatch their enemies as quickly as possible.
Gwen Khan from Outlaw Star has a very eccentric personality, repeating himself, and having a speaking pattern that makes him sound like he's talking to himself. He's also the most brilliant mind in the universe having created both the Outlaw Star and Melfina.
Eiji Kikumaru is child-like, happy, has a flashy playing style and is somewhat of a Cloud Cuckoo Lander as well... but he also has the sharpest vision in the whole team (only matched by Ryoma), is extremely flexible and capable of very high jumps, works as hard as the others to keep himself atop of his team, and is fiercely loyal to his doubles partner Oishi to the point that one of the OVAs and a whole match in the manga are fully dedicated to Eiji's struggle to show Oishi that he's not a burden to him *and* that he wants him back in the team
Princess Tutu has Professor Cat, the main characters' ballet instructor who is literally an anthropomorphic cat. He's shown to be a very competent teacher and is greatly respected by other dancers, but he's obsessed with love and marriage, often threatening his young female students that if they don't keep up with their practice he will have them "MAAAAAARRY ME!" Often followed by him doing something cat-ish like falling to the floor to scratch his back, running into a paper bag, or scratching his claws against a wooden board.
Yomiko Readman (and indeed, almost all Paper Masters) from the Read or Die universe. Deadly with their weapon of choice, and bibliophiles to the extreme. The manga take this to their logical conclusion, as Yomiko and Michelle sometimes get totally physically immersed in their work, allowing the paper from their books to contact their skin...with sexy results.
Minako Aino, a.k.a. Sailor Venus, is the most battle experienced member of the Sailor Team. Which is hard to notice amidst all the ditzy moonstruck behavior she constantly exhibits as well as how she cheerfully murders all kinds of axioms, verbs and phrases. This is downplayed in the live action version, where she is more of an Ill GirlBroken Bird. The manga version of Queen Beryl can testify Sailor Venus is the most competent of the Sailor Senshi, and asks her to stop disemboweling her every time they meet.
Dr. Nii Jienyi carries a stuffed bunny everywhere and babytalks to it, makes completely inappropriate comments to his female assistant, and does not appear to be proficient in social niceties such as regular bathing. He also earned a doctorate at 17, is the only one who knows how to manipulate the Tenchi Kaiden sutras to revive Gyuumaoh, and is the bearer of the Muten sutra and the second-youngest person ever made a sanzo priest.
From Gaiden, Field Marshall Tenpou can't for the life of him keep his office clean enough to find the floor, bathes with a rubber ducky, and walks around in a labcoat and shower sandals rather than military uniform. He's also a Four-Star Badass capable of taking on whole units by himself and a peerless strategic mind.
The sanzo priesthood is essentially an entire religious order of Bunny-Ears Preachers. Because the purpose of a sanzo priest is to protect the sutra they are entrusted with, they are not required to abide by the normal rules of Buddhist monasticism. Every sanzo seen so far has completely ignored the precept of shaving one's head, as well as flouting at least one other precept each. Genjyo Sanzo probably takes this the furthest, as by Word of God the only precept he's (coincidentally) complied with is chastity and he doesn't even personally identify as Buddhist. Because they enjoy extremely high social status, sanzos are also usually given a pass from local social norms. Weird as they may be, however, do not EVER cross a sanzo. They got where they are by being the most competent mofos around.
Amae Koromo from Saki is a monster of a Mahjong player whose bandana looks exactly like bunny ears and her power(?) draws from the full-moon, fitting the bunny theme.
Being eccentric yet ridiculously skilled seems to be a pre-requisite for being in the Shibusen faculty. Of the few staff introduced, only Azusa lacks some odd behaviour. And even she has her crazy map-drawing skills.
Note that this show takes it so far over the top that two of its main trio basically have crippling mental illnesses (one is such an Attention Whore that he literally can't restrain himself from yelling YAHOO at the top of his lungs and leaping atop the most prominent nearby object when he's trying to ambush someone, the other has such Super OCD he's been paralyzed with doubt and suicidal self-loathing because of worries over whether or not he folded the end of the toilet paper roll in his bathroom at home) and yet they're still incredibly powerful combatants… When their act happens to be together.
The bad guys have Crona, a sad, insane, awkward little gender-ambiguous pink-haired kid...who's been fused with the demon sword Ragnarok and turned into the ultimate weapon. At one point they even refer to Crona as "Ragnarok", implying that Crona's presence is merely incidental. And then there's the friendly, excitable werewolf Free, who's depressed after breaking out of a decades-long imprisonment because he didn't escape awesomely enough. He was imprisoned in the first place because of his arsenal of magical abilities and his super agility and strength.
The only reason Excalibur does not count as this trope is that, most powerful weapon in existence or not, his personality quirks manage to drive away even the most willing of prospective wielders. After all, one cannot be a Bunny-Ears Lawyer if one has been unemployed since the 12th century.
Possibly justified as he's one of The Great Old Ones, specificially,the emboidment of Madness Born Out of Anger. However, he is capable of being reasonable and even offered himself to be used, no strings attached when Kid needed to be rescued. He also much more serious back when he was with Arthur, so perhaps it's an eleborate ruse.
She's not good at absolutely everything - she's a godawfully bad filmmaker (due to her phenomenally huge ego which makes her believe that anything she does is automatically Academy Award-worthy). She's pretty hopeless at video games too, due to being too Hot-Blooded to believe in minor trivialities like "strategy".
Dr. Irabu in Trapeze is, save for profession, almost literally a Bunny-Ears Lawyer—but considering his actual occupation the implications are worse; in the anime, he's a psychiatrist who does most of his therapy largely in a psychedelic fursuit (complete with tye-die mascot-esque bear head). The series only gets more insane from there.
Paiway Underberg of Vandread... although it's hard to determine if she's any good at her job, since Duero supplants her as head doctor within the first three episodes.
Also Elsie, airhead low ranked daemon who, nevertheless, is the most successful daemon in the Runaway Spirit Squad. Subverted as she actually depends on Keima and would be useless by her own.
Hajime No Ippo: A rare villainous example in Bryan Hawk. He spends all his time screwing around with women, to the point that it seems the only thing he's interested in apart from boxing, is a gigantic jerkass, has absolutely no social tact and basically does whatever he wants. Oh, and he's lazy. Too lazy to train. And despite all that, he's one of the most fearsome boxers in the entire story. His natural talent is so big that even without training, he easily curbstomps his opponents.
Takamura can also be considered a Bunny-Ears Lawyer, considering how similar he is to Hawk. Tremendous Jerk Ass who pranks people in the shower, has an ego of the size of Japan itself, chases women around and has been featured in newspapers as a huge pervert... and yet he's a brilliant fighter who works very hard for his prowess and wins every fight he's in.
In Deadman Wonderland we have a whole host of really really disturbing Ax-CrazyBlood Knights who are absolutely brilliant at killing people, but who have a lot of really odd... let's just say 'quirks'.
In the manga Category Freaks you have a group of paranormal investigators who are the absolute best at what they do but... er... aren't exactly normal. You have *ahem* a Brilliant but Lazy stand (practically a demi-god), a Camp computer geek, a little girl with bunny ears who eats monsters, a woman whose split personality could kick your ass halfway across the room, and a very polite young man who just happens to spend most of his time as an eye. If you're looking for your sanity you might want to check somewhere three feet north of your brain.
Hinoka from Ame-iro Kouchakan Kandan is a highschool student who is fairly good at construction, as we see with her renovation of the Amber Tea House, but her methods are rather unorthodox. And by unorthodox, I mean Chainsaw Good.
Mr. Shinhue the coroner from The Kurosagi Corpse Delivery Service is a pathologist, mortician, embalmer and historian of ancient Egypt who is an expert on giving customers a pharaoh-style burial (and fairly good at most other forms of burials and respectful body disposal). He also dresses like a mummy, complete with funeral rites in hieroglyphs on the insides of his bandages. While he has severe burns that necessitates wearing bandages 24-7, he seems to like the attire. A throwaway appearance in a later chapter, placed two hundred years in the past, implies that he may actually be a mummy. The story may be non-canon, however.
A double-decker example in Gosick: the detective Grevil acts pretentious and sports a ridiculous hairdo, but the police overlook these things because he solves difficult cases so reliably. Except...he's secretly completely incompetent. To solve his cases (even the simple ones), he relies on a real Bunny-Ears Lawyer: Victorique, an arrogant, manipulative, rolling-around-on-the-floor-to-express-boredom girl who is, unlike Grevil, genuinely very good at deciphering mysteries.
In Penguin Revolution, the president of Peacock has a somewhat idiosyncratic method of running the talent agency; among other things, Ryo Katsuragi is required to attend school disguised as a girl as part of his contract and will be fired of the secret is exposed. He also has the entire agency, agents included, undergo Training from Hell as preparation to put on a play. He can run the agency the way he wants because he's the president, but his methods also get surprisingly good results.
Not surprisingly, a number of the talents employed by Peacock are also rather quirky. The most notable examples are Ayaori Mashiba, a phenomenally talented actor who when not performing is so laid-back as to be nearly comatose, and Yuzuru Narazaki, another highly talented actor who emulates a samurai lifestyle and not only puts himself through voluntary Training from Hell but expects his manager to accompany him.
Tiger & Bunny has Barnaby Brooks Jr., a skilled and popular superhero who is more than a little maladjusted. Apart from never actually bothering to learn any interpersonal skills and suffering from a very apparent case of PTSD, he has a number of odd quirks that range from an aversion to keeping "unnecessary things" (including furniture) to a fondness for suits with an entirely excessive amount of buttons.
There's a lot of Fridge Horror involved here, given how much of Barnaby's personality and actions are a result of Maverick's memory-alteration NEXT power. Keeping him obsessed with catching his parents' killer (re-enforcing the PTSD) and ensuring he didn't develop significant relationships with anyone, Maverick had spent several years abusing Barnaby's unconditional trust in him to keep the young man engaged in his work-driven lifestyle, which helped to further Maverick's goals in making HeroTV a more profitable enterprise.
Several of the others count: Sky High is a very successful hero in spite of being quite the Ditz; Origami Cyclone's fanboyism (which is actually his selling point) belies his actual talents (not helped by Ivan's lack of confidence). You also wouldn't expect that the showy, flamboyantly camp Fire Emblem owns the company he's affiliated with, meaning Nathan must be pretty successful to stay associated with HeroTV.
Just about everyone in Kenichi: The Mightiest Disciple, but the best is Siegfried. He'll come across as a crazy, and sometimes scary martial artist, who won't stop getting up after getting the crap beat out of him. After getting to know him, he's less scary, but he's downright weird. He'll fight with anyone for inspiration for music, and sometimes use their bodies as sheet music. That being said, he's still one of the most powerful members of the Shinpaku Alliance.
The Phantomhive Staff from Black Butler are hopelessly incompetent at their service jobs. But cooking, gardening, & cleaning aren't what they were REALLY hired for, as they are top notch assassins. Also Grell as a Shinigami, but ONLY in the manga.
Also Chapter 66 of the manga had a significant portion of the cast wearing literal bunny ears. (The reason is that it was an easter special)
Goldfish Warning: inverted with Tanaka, the worst lawyer/butler ever, who somehow remains employed by his former client's daughter's rival, & literally wears bunny ears on a several occasions.
In High School DXD, Rias describes the four lords of hell as this in their private lives and later on, the other leaders of the different factions are this and it shows. Serafall loves to cosplay as a Magical Girl, Sirzechs and Azazel made an entire Super Sentai TV series based on Issei and Rias after Issei achieves Balance Breaker by pushing Rias' nipples, and Odin visits love motels just to name a few.
Date Masamune from Gate 7. He's a Bratty Half-Pint who doesn't like to be reminded of that, wants to be an adult quickly and therefore he lies saying that he's in 6th grade since he was in 1st grade; he's acts as the typical in-love-teenager girl towards Hana to the point to imaginate her as a Cat Girl at the middle of a serious fight and it's even Lampshaded that he is comic relief once. Who would believe that this boy is really a excellent swordfighter and an honors student?
Ms. Yoshinoya from Hidamari Sketch is immatureCosplay Otaku Girl who also like to make sexual statements to her students. She's still at the job exactly because the Principal recognized she's an example of this trope—she's just great in teaching art.
Misaki, a hyperactive, obnoxious Genki Girlotaku that nearly single-handedly made an anime, sold it, and was commercially successful;
Jin, a Cananova (out of an inferiority complex towards Misaki) who wrote the script for Misaki's anime; and
Ryuunosuke, a hikikomori programmer who already has contracts with major corporations.
Harley from Pokémon, as well as his DP-era Expy, Nando. The two of them act extremely feminine (to the point of being Ambiguously Gay/Camp Straight), Harley obsesses over his cooking and Nando dresses like he lives in the 15th or so century, yet they both manage to be very skilled Coordinators, reaching the Grand Festival easily.
Manabe from Kotoura-san may be kooky as all hell, but he constantly uses his zaniness as a calculated method of reassurance or encouragement for the former Broken Bird Haruka exactly when she needs it.
Many of the senior officers from the Survey Corps, in Attack on Titan. It's even pointed out that their bizarre quirks are tolerated or humored solely because they are so good at their job. Captain Levi is a Super OCD clean freak.. but he's also the World's Strongest Man. Squad Leader Hange Zoë is a would-be Fluffy Tamer, but also an exceptionally capable (if slightly mad) scientist and field commander. Mike Zacharius has a habit of smelling people as a greeting, but Eren is told to ignore it because Mike is one of the best soldiers in the military.
Nyarko herself may qualify, considering she's extremely goofy, lazy, and capricious, and yet never loses her job with the Space Police because she's apparently a well-respected and successful officer. This may be an Informed Attribute, or it could be that meeting Mahiro and becoming utterly devoted to bedding and wedding him (in that order) has taken all her focus.
The titular character of Puella Magi Kazumi Magica comes off as this. Among other things, she reads a person's morality by how they eat, demands that everyone she trains shout their attacks in Gratuitous Italian, and claims that her ahoge can detect witches. She also happens to be the ace-in-the-hole of the Pleiades Saints, and has taken down Walpurgisnacht-esque entities almost singlehandedly.
Teekyu has in episode 33 a priest who dresses in gyaru fashion, with bright clothes, an extravagant hairstyle and lots of make-up, organises a funeral with the atmosphere of a party, has her phone beeping during the ceremony, and stumbles around drunk afterwards... then recognises instantly when the deceased's spirit has taken over someone's body and flushes him out, making the protagonist acknowledge that she deserves her position.
Minori Koganuma of Outbreak Company is initially shown to be a serious member of the JSDF, but she fits this trope, thanks in no small part to her being a huge Yaoi Fangirl.
In Yu-Gi-Oh! 5Ds, there's Akatsu, a nutty, spacey, and exuberant fellow, who's also the genius behind the current Momentum Reactor. (Of course, given what it can potentially do, this attitude may not be a good thing...)
Saijou no Meii: Although Mikoto can be very childish at times and is obsessed with collecting Mushiman merchandise, he's extremely competant at his job as a pediatric surgeon.
Graham from Mother Keeper, as a killer he's fantastic and he's fully capable of getting past the greatest security in Eden without any hassle. Yet he has no clue as to how to manage money, how to tell quality from a cheap rip off, his plans often end in someone getting killed, he tends to go overboard when killing and he forgets to take feelings into account.