Two very different characters are obliged to work together to solve a crime. The premise is very often, but not always, expressed in trailers and pitches as "She's an X. He's a Y. Together, they fight crime!" So much that it's almost become a Stock Phrase. Expect lots of cultural clash humor arising both from different policing traditions and from general personal differences between the two characters. Sometimes the ante will be raised by making the crime-fighters from two groups that are actively antagonistic.
The story often (but not always) involves one character being on his own turf while the other character is a visitor. If there is a sequel, the character who was on his home turf in the first story will probably be visiting the home turf of the other character (the reverse situation).
Can be the premise of a Buddy Cop Show, although then there usually needs to be some reason for the other cop to continue hanging around beyond the pilot.
They Fight Crimemust use both Wunza Plot and Odd Couple. Please do not add teams of more than two characters, or single characters, no matter how quirky they are.
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He's a weed loving gangsta rapper. They're a large respected computer security company. They Fight (cyber)Crime! Seriously.
Anime and Manga
Majin Tantei Nougami Neuro: He's a demon from hell with an insatiable appetite for MYSTERY, she's a high school student with an insatiable appetite for FOOD. They fight crime!
Detective Conan: He's an age-regressed Sherlockian super-sleuth, she's his unwitting girlfriend/baby-sitter. They Solve Crime!
Every time Togusa is teamed up with Batou in Ghost in the Shell, their relationship fits this trope quite closely. While they have been working in the same unit for quite some time, Togusa is a cautious and serious family man who used to be a police officer, and Batou a former special forces soldier who regularly leaves a trail of destruction and badly beaten up suspects.
Power Man and Iron Fist: Luke Cage is an indestructible badass from Harlem. Danny Rand is a martial arts master CEO. They Fight Crime!
Cable is a telekinetic cyborg soldier from the future with a messiah complex. Deadpool is an insane mercenary who's in love with Bea Arthur and Chimichangas (and Squirrel Girl). They Fight Crime! Or cause it, it shifts.
Spider-Man had an issue where Mary Jane helps proposes the following TV pitch: "She's an actress. He's a super-hero. They solve mysteries. Think it'll fly?"
Last Action Hero: In the Parody-riffic Schwarzenegger vehicle, the audience gets a peek into how partners are assigned at the precinct. "Kowalski, you get the rabbi. Jenkins, you have the animated cat this week."
The Millennium Trilogy: He's a middle-aged journalist and chick magnet. She's a twentysomething bisexual ass-kicking computer genius who may have Asperger's syndrome. They Fight Crime!
Catopolis: Two household pets gain psychic powers, and one of them lampshades it.
Huey:You're an Oracular Cat, I'm an Oracular Hamster. We fight crime!
Most any of the partnerships in the Lensman series, but in particular: One's the top agent of humanity. The other is a thirty-foot-long winged reptilian alien with multiple eyes on stalks. They Fight Boskone!
Past Life: He's an ex-cop whose wife died (who is totally not going to come up in the series eventually), she's a past life therapist. They fight reincarnation-based crime!
Big Wolf on Campus: He's a perky, easily frightened, goth with an encyclopedic knowledge of the supernatural; he's an easy going, football playing, werewolf with a secret identity and the ability to kick lots of ass. Together, they fight Monsters! With a Kickboxing, Totally Bad Ass Normal, Action Girl.
Torchwood, by virtue of being a police procedural. He's an omnisexual reformed conman who can't stay dead. She's a tough-as-nails policewoman who keeps him sane. They fight crime!
The Sweeney: He's a tough, no-nonsense, bends-the-rules older cop who's quick-witted and with an eye for drink and the ladies! He's a younger, more inexperienced cop who wants to follow the rulebook yet respects his older partner! They fight crime!
Keen Eddie: One's a scruffy, accident-prone, idealistic detective from Manhattan, the other's a shallow London dandy with an addictive personality and an over-active libido. Together, they fight crime!
Dark Angel: She's a genetically-engineered super-soldier Half-Human Hybrid. He's an incredibly wealthy paraplegic who spends his spare time as an airwave-pirating cyberjournalist. Together, They Fight Crime! Although in season 2, she frees more super-soldiers and they stop fighting crime so much.
White Collar: He's a blue-eyed master criminal genius with a million dollar smile. He's a no-nonsense FBI agent who always gets the bad guys. They Fight Crime!
Dexter: He's a mild-mannered lab tech with a Dark Passenger. She's a tough, foul-mouthed cop with a heart of gold and a lot to prove. Together, they fight crime (in their own special ways)!
Supernatural: Two brothers—He's a pre-law student who just lost his girlfriend. He's a charming blue collar bum who can't find their dad. They fight monsters!
The Invisible Man: He's a recently reformed thief with an invisibility gland in his head. He's a jaded veteran and paranoid conspiracy theorist who's been fired from every other government agency in existence. They fight crime!
Campion. He's a disinherited member of the aristocracy. He's a former burglar turned valet. They fight crime!
In "A Good Man Goes To War" and "The Snowmen" we have: She's a lesbian Victorian silurain with katanas; she's her human butler and lover, and he is a gun toting potato dwarf. Together, they fight crime! (Including eating Jack the Ripper)
And from the original series, we have...
...he's a robot dog...
...she's a sexy jungle girl from outer space...
...he's a kilt-wearing 18th century Scotsman...
...they're his granddaughter's science and history teachers...
...he's a teenage supermathematician from another universe...
...she's a teenage girl who likes to make things explode...
...he's an obnoxious schoolboy in thrall to the universal embodiment of chaos...
Common Law: One's a street-wise loudmouth with a lengthy list of disgruntled ex-girlfriends. The other is a logical former lawyer who's still in love with his ex-wife. They Fight Crime!
Arrow: One's a playboy socialite who was trapped on an island for five years, returning to fix the mitakes of his father and save his city. The other's his bodyguard, a former marine looking for a way to help the world and fight for a cause worth fighting. They Fight Crime, corruption, and the occasional assassin!
Person of Interest: He's a former spy disowned by the CIA. He's an independently wealthy inventor with a machine that spies on everyone. Together, They Prevent Crime!
This seems to be the premise of the new Hannibal series: he's an empath FBI agent, he's a cannibal psychologist. They hunt serial killers! And get increasingly and quietly drawn into madness/increasingly and quietly draw into madness, respectively.
Paper Mario The Thousand Year Door referenced this trope during the Glitz Pit challenge. When Mario converses with a Spike Top at one point in the rankings, he talks about distinctive features, his being his spike and Mario's being his Badass Mustache. "Spikey Joe and Mustache: Together, they fight crime! That'd make for a killer TV show!"
On the other hand; He's the aforementioned laid-back amnesiac ghost. He's the ghost of a hyperactive puppy pomeranian who will do anything to protect his little miss. They fight crime!
This one gets even better once you beat the game and find out... that you were a cat all along!
One's a serious, idealistic (and quite corpulent) detective who goes by the book and strives to give the best out of himself. The other is a stylish, cool-cat (and quite lanky) police inspector whose every move is literally a dance move and still climbs up the ranks because he's simply that good. They are best pals, and fight crime!
One's said lanky stylish police inspector, the other is a grumpy former forensic medic for the police who left his job to research a meteorite. They chase down vengeful ghosts!
In The Curse Of Monkey Island, there is a book about ventriloquism which, although only needed for one puzzle, can be used on numerous objects and characters with amusing results. Using it on the head of a donkey costume in the local theater results in, "Hi. I'm Frank. I'm a talking donkey. I fight crime."
He's an amnesiac cybernetic soldier who doesn't know what to do! She's a megalomaniacal artificial intelligence dominatrix who's lost all their power! They Fight The Many! You and her versus the world. She really doesn't like that.
"He's a short-sighted alcoholic vampire hunter with a passion for fast cars. She's a manipulative African-American Valkyrie on the trail of a serial killer. They fight crime!"
"He's a benighted crooked firefighter plagued by the memory of his family's brutal murder. She's a supernatural gypsy research scientist who hides her beauty behind a pair of thick-framed spectacles. They fight crime!"
"He's a time-tossed bohemian waffle chef with a robot buddy named Sparky. She's an artistic mutant museum curator who hides her beauty behind a pair of thick-framed spectacles. They fight crime!"
"He's an otherworldly shark-wrestling librarian in a wheelchair. She's a disco-crazy mute mechanic from aristocratic European stock. They fight crime!"
"He's an otherworldly one-eyed master criminal from the 'hood. She's a mistrustful tempestuous socialite living homeless in New York's sewers. They fight crime!"
"He's a fast talking gay dog-catcher haunted by memories of 'Nam. She's an elegant goth vampire looking for love in all the wrong places. They fight crime!"
"HE'S An Inquisitor held back from promotion due to his radical ideas of fairness, equitability, and investigating before Exterminating! HE'S a two-legged biological killing machine which consumes men to gain their skills! Together THEY FIGHT HERESY."
Blamimation: He's a Renaissance trout! He's a modern-day hawk! They fight crime! But only above water...
An episode of Family Guy played this as a joke, with the newest cop show, where the hardened cop got paired with a kid from the streets, a robot, and a couch To Fight Crime!
Parodied on The Critic, when Clint Eastwood played a cop partnered with a woman, a cute little kid, an ugly old dog, a dinosaur, and a leprechaun. Later, Arnold Schwarzenegger shows up partnered with a pig, an alien, Siamese twins, a sofa, and a second-rate mime.
"A fugitive, semi-aquatic special forces amateur stage magician, framed for a crime he didn't commit, the 1865 assassination of Abraham Lincoln, joins forces with a rogue-trillionaire inventor-extreme fighting champion from the future. Together with the aid of "R.I.C.K.", their super crime-fighting talking high-tech rickshaw, they'll bring hope, justice, and varying degrees of aquaticness to a Tri-State Area in peril. Together, they are...Doof 'n' Puss!"
Gargoyles: She's an NYPD detective. They are winged beasts of the night who spent the last thousand years in an enchanted sleep. They fight crime.