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"She's a wiccan! She's a nun! Can two sisters share an apartment? Find out in Which is Witch?!"
—Observer, Mystery Science Theater 3000, parodies this formula
Also known as One's A Plot.
Many TV shows are based around a High Concept -- a simple, intriguing premise that can be explained in one or two sentences -- a "hook" which attracts the interest of everyone from producers to advertisers to viewers. One very common way to get a build a high concept is to take two characters who are very different from each other, quirky, or somehow odd, and then team them up to accomplish some objective. In 'the Biz' this is called a "Wunza plot". Put in its simplest form, a Wunza Plot is "One's a [X]; one's a [Y]. Together they [Objective Z]" 
Oddly, even the most bizarre premises can end up rather mundane in execution, with the strange characters -- despite initial weirdness -- ending up in fairly stock show formats like workplace comedies/dramas, quirky sitcom family hijinks, or, yes, fighting crime as private detectives or freelance do-gooders.
Animation obviously has great potential for silliness, especially if it's an Animated Adaptation of a live-action show.
A silly premise isn't always fatal, and the mention of a series below is not necessarily a criticism. With the right casting and writing, some shows which sound totally ridiculous have been hugely successful and even critically acclaimed.
Anime and Manga
- Saint Young Men: Buddha and Jesus share and apartment in Japan. They do ordinary day-to-day stuff.
- Tiger and Bunny. A down-on-his-luck, goofy single father and an icy-cold pretty-boy upstart are paired together as a corporate-sponsored superhero duo. They Fight Crime.
- The premise of Dirty Pair in which the impulsive tomboyish Kei is paired up with the more cerebral and mannered Yuri. Of course it's all right since They Fight Crime...
- Referenced in The Sandman: Matthew (a servant of Morpheus who happens to be a raven who used to be a man) describes his brief adventure with the Corinthian (a serial killer with fanged mouths in place of eyes) as being "like a bad TV show." Borders on a Lampshade Hanging, except that unlike most of these examples, the weird team-up was only a small part of a much larger story.
- Angel and the Ape ('60s comic book with revivals in the '80s and '00s): She's a gorgeous super-sleuth who speaks thirteen languages and knows karate. He's a comic book artist who happens to be a talking gorilla. They Fight Crime!
- Sam and Max: One's a dog. The other's a hyperkinetic lagomorph. They Fight Crime
- G.I. Robot., he's a robot. His buddy's a hard bitten G.I. They fight dinosaurs and Japanese giant robots until the robot is sent into space with military werewolves, vampires, etc. and the comic's creator. Bob Kanigher, man..
- Duplicity: On its movie poster - "She's ex-Cia. He's ex-MI 5, together they are stealing a fortune." She is Julia Roberts, he is Clive Owen.
- Parodied in Network: "These are those four outlines submitted by Universal for an hour series. You needn't bother to read them; I'll tell them to you. The first one is set at a large Eastern law school, presumably Harvard. The series is irresistibly entitled "The New Lawyers." The running characters are a crusty-but-benign ex-Supreme Court justice, presumably Oliver Wendell Holmes by way of Dr. Zorba; there's a beautiful girl graduate student; and the local district attorney who is brilliant and sometimes cuts corners. The second one is called "The Amazon Squad." The running characters include a crusty-but-benign police lieutenant who's always getting heat from the commissioner; a hard-nosed, hard-drinking detective who thinks women belong in the kitchen; and the brilliant and beautiful young girl cop who's fighting the feminist battle on the force. Up next is another one of those investigative reporter shows. A crusty-but-benign managing editor who's always gett..."
- Steel Justice: see for yourself. Twenty Minutes Into the Future a cop loses his son to a car bomb and the boy is reincarnated as a toy robot dragon. That can shapeshift into a giant robot dragon and help his dad bring his killers to justice. Apparently the purpose was as an extended advert for the toy dragon in question. Perhaps the best bit is that this truly outlandish premise is paired with one of the blandest, most generic titles imaginable.
- The Sookie Stackhouse Mysteries: She's a luscious, highly desirable telepathic barmaid. He's a vampire suffering from anachronistic attitudes.
- Magic Pickle. He's a super-powered pickle. She's a little girl. Together they fight super-powered evil produce.
Live Action TV
- Friends featured the example of 'Mac & Cheese' (Joey's starring role)- one's a hard-bitten detective, one's a robot. They fight crime.
- Blood Ties: She's a Canadian police detective who is slowly losing her sight. He's a vampire. They fight crime!
- Psych: A devious young man with uncanny detective skills pretends to be psychic in order to solve mysteries. With his nerdy, put-upon Black Best Friend.
- 'Being Human: A nerdy werewolf, a charismatic vampire and a sexy ghost are roomates. They try to have a normal life.
- Constantly parodied in Mystery Science Theater 3000 when reading the title credits for the movies they watch. e.g. Tom: "Body Care and Grooming! They're cops!"
- The Middleman Superhero with android secretary hires art student with crazed activist roommate as Sidekick and heir apparent.
- Heat Vision and Jack was a show pitched by Ben Stiller whose pilot (the only episode ever made) is a bit of a cult classic that parodies this whole trope. The premise is that a former astronaut with solar powered super genius (Jack Black) and a talking motorcycle (voiced by Owen Wilson) cruise the country as drifters on the run from NASA. They Fight Crime!
- Whose Line Is It Anyway? had a game revolving around this trope. The performers had to act/sing the opening sequence of a comedy whose title was supplied by the audience. Some of the fictional comedies:
- This Morning with Richard Not Judy mercilessly parodied this often. Such as this example:
Ian Roll is a driving instructor, Ian Reversal is a baker. Due to some kind of accident they are forced to swap jobs. With hilarious consequences.
- Or this one:
Ian and Iain Bent are brothers who are policemen. One is corrupt and the other is homosexual. They both suffer from curvature of the spine, and they're made of copper - they're robots in the future.
- Dark Angel doesn't fall under this trope so much as leap into it headfirst from the roof of the Space Needle:
She was designed to be the perfect soldier. She was trained to be a human weapon. But then she escaped. They came after her, and she knew they would never stop looking. [...] She's found an unlikely ally in Logan Cale. Born to a life of privilege, he's now an underground cyber-journalist crusading against a corrupt government. He wants to save the world, or what's left of it. She just wants to find the others like her. Together... who knows?
- Life On Mars: He's a cop from The Noughties. He's a cop from The Seventies. They fi-- ...oh.
- Scarecrow and Mrs. King': He's a spy; she's an everyday housewife. Together, they ...do spy-ish things.
- BJ And The Bear: Wunza hunky truck driver, wunza chimpanzee. Together they solve other people's problems.
- Alan Partridge follows this trope faithfully with the majority of the ideas he pitches to a senior BBC executive for new shows.
- Metal Wolf Chaos: He's the President of the United States, she's his secretary. They fight The Evil Army and the Vice President using a Humongous Mecha and Burning American Freedom.
Cop: You're alright, Pony. You're alright.
- Agents of Cracked. One's a mentally-unfortunate, possibly magic, pansexual judge with hammerspace, the other is... pretty much Charlie Brown. Together, They Write Jokes!
- Kim Possible. She is the popular cheerleader. He is the under achiever who has no friends. Together they fight crime, and she'd better be home before 10.
- The Sylvester and Tweety Mysteries.
- They're Pinky and The Brain. One is a genius. The other's insane. To prove their mousey worth, they overthrow the Earth.
- They Fight Crime! creates these via Mad Libs.
- Hence, the name: "One's a" = "Wunza"