A truly astounding example: Dr. McNinja. He's a doctor, and a ninja (and constantly masked, to boot). That alone should have been enough, as each profession is the bunny ears for the other. But he's also obsessed with Batman, plus everything else about him.
Dr. McNinja is self-employed. Judy, on the other hand, is a good receptionist who happens to be a gorilla.
Danny Hua in Nukees is a sweet-tempered Cloudcuckoolander who can't follow most basic metaphors, is compulsively honest and has a phobia of speaking with contractions. He is also a brilliant nuclear engineer and one of the finest weapon designers in the world, being personally responsible for the creation of the Giant Robot Ant. There currently is a small contingent of groupies who have decided he is the Wise One for his incomprehensible, unintentionally profound-seeming speech quirks. He is also, according to author Darren Bleuel, based on a real friend of his, and there is a page full of quotes to prove it!
What about King Luca? He literally believes himself to be royalty, wears a crown and cloak, and routinely grants land, nobility, or knighthood to his followers. And he does have followers, because he's a wonderful physics tutor who singlehandedly helped many students pass their classes.
And Lord Shojo, who pretends to be senile in order to ward off assassination attempts, but was able to fool not only the paladins he controls, but also the entire Order of the Stick, despite not having any levels in a PC class.
In Sluggy Freelance recruiting Bunny Ears Mad Scientists (but keeping them at a safe distance) is basically the whole point of the Hereti Corp freelancer program.
Doug from Fletcher Apts. In high school, he was head of the Science Honor Society and #2 player in the chess club. In college, he's the vice president of the psychology club. Also, he's known for barfing easily and freaking out over female breasts.
Taisei from Sakana is a very goodhearted, yet silly and naive person who gets drunk often and is a walking source of Ho Yay. He's also a successful assistent director for a pretty big company and the smartest man of the whole cast.
Rayne Summers from Least I Could Do. Marcy McKean - Rayne's boss and CEO of the company - puts up with his Red Bull addiction, childlike behaviour, and non-existant attention span because he is genuinely good at his job.
Othar is a "Spark", and Sparks are the incarnation of this trope and Mad Scientist turned Up to Eleven. They're all highly competent geniuses and supergeniuses; and are all completely and totally barking mad.
Collar 6: Sixx is the very successful owner of a chain of hotels, so no one really cares that she has a harem of hundreds of (willing) lesbian slave girls.
But, considering the background material on the world the author has recently been providing, this is not considered to be in any way unusual ... in fact, it's considered rather admirable.
Blade Bunny is a bunny ears assassin. A lot of people don't take her seriously because of her style, but then they see her in action.
Darkly subverted in Jack. Doctor Thalmus thinks that developing a cure for cancer makes his crime of pedophilia acceptable, even going so far as to hold the cure hostage when Aurthor finds out. No one else seems to agree with his viewpoint.
MSF High. In an example that counteracts Doctor McNinja...the entire staff. The sanest people are the Mad Scientist who is addicted to coffee, and the Vampire with a harem. Name a staff member who is not certifiably insane, and frighteningly competant.
Sarabeth Phoenix of Wright as Rayne is the fairly serious head of both a company and a criminal group, but is big on anime and dyes her hair blue.
Gatemaster of Survivor: Fan Characters is a Cloudcuckoolander who considers his alliance a group of 'friends', and is perhaps one of the most competent strategists in the game, managing the orchestrate (through a combination of blind luck and coincidence), the final four of his choosing. And he does this without even realising he is in charge.
Homestuck: oddly enough, the many variants of the Courtyard Droll, who is completely unprofessional, willingly dresses in the most ridiculous outfits possible (enormous and bizarre hat completely mandatory), has a silly happy dance that he pulls out at the slightest opportunity, and hatches absurd plans involving things like wearing C4 on his head, smuggling a pumpkin full of knives into a prison cell, or constructing a bomb out of shaving cream...which work. Next to his boss, he was one of the quickest members of his gang to actually rack up a body-count.
A number of the characters in Grrl Power fall under this, from Math (lecherous but unspeakably talented martial artist), to Heatwave (capable of sublimating tanks, but dumb as a brick), to Sydney (has a suite of eight orbs that give her unique and strong powers along with comic book knowledge to apply those powers properly, but severe ADHD, trouble with authority, and overall a PR nightmare). This is explicitly justified; powers are rare enough that they can't pick and choose their team. As long as a super doesn't have an actual criminal record (and possibly sometimes even then), they're invited to join, since it's better to have them on the team where they can be monitored and trained than off it where they can get into trouble.