Jughead Jones of Archie Comics fame is sometimes portrayed as this, though it Depends On The Writer. One issue even had Mr. Weatherbee and Ms. Grundy furious at the revelation that Jughead was getting straight-As despite being such an oddball and slacker, and utterly refused to believe it was due to anything other than cheating until he aced a pop quiz.
Mr. Weatherbee Let's just face facts. His mind is a steel trap. Whatever he hears he never forgets; he just files it away until it's needed. He's a sloppy, unkempt, irritating genius.
He even intentionally got himself sent to detention so he'd have a quiet place to study, it never occurred to him to just ask if he could stay after class.
Julius Caesar in Asterix is irritatingly pretentious, has various odd traits and is weirdly fixated on conquering a tiny piece of valueless territory belonging to a bunch of losers and madmen, who he is nonetheless Friendly Enemies with. He's also one of the greatest military leaders ever to live and by far the cleverest, noblest Roman we're ever introduced to. Cleopatra was also like this — an excellent queen but also a materialistic spoiled brat who eats disgusting food — but she mellows out as the series progresses.
Beast Wars: Uprising: As in Robots in Disguise, Bisk believes he's a video-game character. And yet he's skilled enough that Megatron hires him repeatedly for jobs that need doing, and he pulls them off, such as sneaking into the most heavily fortified base on Cybertron in a cardboard box, and figuring out where to go solely from a random bit of chatter overheard from guards. He also manages to survive a Zombie Apocalypse, figuring it's just some new content patch, having the time of his life as he does.
Blue Beetle: Ted Kord is an immature nerd who never stops telling stupid jokes, dresses like a kid's show host, and has a disorganized lab. He's also a technological genius, expert fighter and detective who uncovered a conspiracy even Batman missed and died trying to stop it.
James from Bowling King is a former champion bowler with incredibly formidable skills. Not that his protege Shautieh could tell after spending an entire morning of shopping with him; all James seems to care about is his wife, and he rarely focuses on what he's actually supposed to be doing... but he still manages to see patterns and method to Shautieh's approach that even he wasn't aware of before.
Dark Reign: Made Men is a one-shot made up entirely of stories about Norman Osborn and HAMMER making deals with guys like this. Bringing them back to life, getting them out of prison, that sort of thing. Spymaster is a peerless industrial spy and thief... and a con artist who is practically afraid of truth. Attuma is a fearsome and powerful warrior... and a rage case who doesn't understand anything but conflict. The Gamma Corps are an elite fighting squadron of Gamma-enhanced soldiers... and extremely misguided crusaders with a warped sense of justice.
Deadpool could very easily fit into this category as, despite his blatant insanity, tendency to annoy others and, occasionally, sheer stupidity he is frequently hired as he's just too good at his job.
He also strays into Crazy Is Cool sometimes; for example, when he fights with absolutely no semblance of sense, not even deciding what an attack is going to be until it connects, to beat the Taskmaster.
However, Taskmaster himself called Deadpool out for this, telling him that one of the reasons he's not taken seriously and doesn't make nearly as much money as he could is because he's so crazy that it's even more crazy for people to think they can rely on him. Hell, in some of his earliest incarnations, he would change sides in the middle of a fight if he was offered more money. If you have a hopeless suicide mission, call Deadpool. If you need subtlety and discretion, stay light-years away from him. Also, since Deadpool has developed a conscience and tries to do the right thing, there's no guarantee that he won't turn on someone who just paid him if he thinks the situation warrants it.
Fantastic Four: Victor Von Doom is one of the foremost scientific minds in the world, possibly the universe, who has elevated his home country of Latveria from struggling failed state to fully functioning global power. He just also happens to be a narcissistic, paranoid, operatic despot who goes around in Powered Armor he made, occasionally trying to kill a man he went to college with for something that wasn't the man's fault. Any time he's deposed from ruling Latveria, the alternatives always turn out to be far, far worse.
Raptor in G.I. Joe comics. Cobra's accountant, he's a Mad FALCONER who dresses like a bird at all times and is regarded as a complete nutjob even by other Cobra operatives and Cobra Commander himself. On the other hand... he's an okay accountant and nobody else would work for Cobra.
As various deconstructions and parodies have shown, virtually all superheroes are Bunny Ears Lawyers (more so during The Golden Age of Comic Books). Seriously, dressing up in bright colors and knocking people around can only be accepted when you're really good at what you do.
Orange Crows: Although the Special Witch and Warlock Squad, or S.W.S., is fully staffed by quirky, loose-cannon teens and early-20-somethings, Bianka is the only real Cloudcuckoolander of the bunch. She's a bratty teen who's declared herself the "Queen of Cats," and has plans to build a 1,000 lb "teapot tank" to rain porcelain destruction down on her enemies. That being said, she's a talented Witch with impressive reserves of mana (as pointed out by her supervisor Natalie), naturally gifted at high level magic (like making animals talk), an accomplished athlete, and capable of casting powerful offensive spells when in combat.
In Paperinik New Adventures there is an actual lawyer, Eugene Photomas, that constantly forgets dates, places, and names (even his own, at one point). However when he is in a courtroom his competence switch is flipped, to the point to be considered one of the best lawyers of XXIII century.
She-Hulk is sometimes a literal example (literally a lawyer, not literally Bunny-Eared). She's also green, incredibly muscular and sometimes a superhero on the side. (And often works for a Bunny-Eared Law Firm.)
Even Spidey himself is a Bunny Ears Lawyer amongst the superheroes. He's known as being goofy and weird on top of being a hugesmartass but most heroes consider him a brave, versatile and valuable asset amongst the numerous street levelers. Being the main hero of the New York Streets helps. Also he's a lot smarter than he looks and possibly the most teamed up hero in the Marvel Universe of his power level.
Before he became Electro, Max Dillon was so greedy he wouldn't do anything unless he was paid. He only saved the life of a coworker after being offered a large bonus. He was only tolerated because he was the best pole man in New York.
Famed reporter Clark Kent, a.k.a. Superman, has a horrible attendance problem. Tolerated, because he brings in the good stories anyway, and one of the reasons Clark chose "reporter" as a job in the first place was flexible hours. It doesn't hurt that Clark happens to be the fastest typist on staff.
Lois Lane is a Pulitzer prize winning journalist, who depending on the continuity has the self-preservation skills of a suicidal lemming, mad obsessive streaks (at least in the Silver Age), has been known to schedule interviews with Superman by throwing herself off buildings, and on occasion having a surprisingly shocking grasp of basic spelling.
In one story, Superman assumes the identity of Johnny Clark, a firefighter. His taciturn standoffish nature is tolerated by the Metropolis Fire Brigade because he's just that good at saving people. The fact that this new identity allows him to help people 24/7 could also mean the Johnny Clark persona is a Bunny Ears Lawyer for Superman as well.
In Superman Smashes the Klan, Clark is treated like this at the Daily Planet. On one hand, he's an excellent writer who always finds the scoop. On the other, he has a habit of vanishing at inconvenient times and does odd things like sniff himself for the smell of alien crystal.
Arthur on The Tick is a perfectly normal everyday human and an experienced chartered accountant in an accountancy firm... Who also wears a full-body moth suit. Even to work. In an aversion, this gets him fired in all continuities.
Arthur: I'm fired? Is there a problem with my work? Mr. Wiederspan: No, no, no, no, no. It's just... Um... IT'S THAT STUPID BUNNY OUTFIT!
The Tick, meanwhile, is trusted by his community as a superhero and protector of the innocent (and not, say, carted off to the nearest insane asylum) in spite of his... Eccentricities. This was actually averted in the comics continuity, in which The Tick started out as an asylum inmate before escaping and becoming a superhero.
Tintin gives us Professor Cuthbert Calculus. He's prone to mishear every single word being said to him due to his hearing impairments and he's a borderline Mad Scientist. However, he so brilliant that he kept getting hired by various groups and governments to design various inventions or to help their science programs.
Professor Calculus was actually the latest in a list of kooky scientists, who apparently held considerable respect in their fields despite their silly behavior (key of which has to Professor Phostle, who discovers a meteorite is apparently going to smash into Earth and can only think of the fame he'll get as its discoverer). He was such a Bunny-Ears Lawyer, however, that he broke out and became a regular part of the cast.
Alexander Glushko of Top 10 is a darker take on the trope, since his bunny-ears are rampant alcoholism. However, since he is the only telepath the Neopolis PD has access to, they're willing to overlook him being black-out drunk on the job as long as he is capable of telling interrogators what suspects are thinking.
The Transformers: More than Meets the Eye is set on a ship full of people who are, for one reason or another, utterly dysfunctional, but enough of them are good at their jobs that they can get away with only moderate uselessness. Brainstorm is a Mad Scientistand double agent but a genius who can build basically anything no matter how impossible; Whirl and Cyclonus are Jerkasses most of the time, but are both talented warriors even though they spend most of the first season despising each other (Whirl also, at one point, successfully performs robotic open-heart surgery out of spite by repurposing his knowledge of watchmaking); Ultra Magnus is so inflexible about the rules he has been known to consider people suspicious for wearing their badges at an angle, but backs this up with incredible strength and military skill; Chromedome is a Death Seeker and a brilliant mnemosurgeon. Even the command staff get in on this: Rodimus is a hotheaded Glory Hound who has been known to pretend to be dead to avoid difficult questions, but he's also the only one actively pursuing the Knights of Cybertron and is a skilled leader and warrior when he can be bothered, and Megatron is freaking Megatron.
The Scavengers are even worse, but they do have some skills — particularly Spinister, who is a medical genius, and also an utter lunatic who classifies random objects as "goodies" or "baddies" depending on the noise they make when he hits them.
Krok: Spinister's a surgeon — an exceptional one. Don't be fooled by his behaviour, his attention span, or his inability to differentiate between day and night.
Transmetropolitan's Spider Jerusalem is a drug addict, a sociopath, a pervert, a world class cynic who despises his own audience...and a damn fine journalist.
Yelena: It's like working with a four-year-old boy with a massive caffeine high and a permanent and yet very small erection. And you know what the worst thing about it is? He's the good guy.
