American Dad doesn't have the best reputation, as it was created by Seth Macfarlane and began its life as a shallow Family Guy clone (yes - shallower than Family Guy, I said it). But the show morphed into something weird and unique and - gasp - pretty good by its third season, bringing in weird running gags that would gradually spiral out of control. One of these gags were teenager Steve Smith and alien bud Roger pretending to be cops dubbed "Wheels & the Legman" ("Wheels" because he had to rely on a wheelchair, and "The Legman" because...he didn't). What began as a parody of late 70s cop shows soon morphed into the two actually solving cases, although almost every time the culprit ended up being Roger.
Have you seen the movie Constantine, starring Keanu Reeves as John Constantine? Good! Forget you ever saw that! Constantine (in the comics) is a blonde Brit focal point of the series Hellblazer, about a guy who has the MOST SINFUL BLAZER IN MEN'S WEARHOUSE HISTORY! Oh wait actually no it's about a chain smoking, cynical, occasionally embittered detective of the occult and magician (not the kids' party kind, unless you wanna give your kid the worst birthday ever).
Max Payne, also known as the Worst Pun-Named Character In Videogame History, has had it rough - pretty much everyone knows and/or cares about dies, he frequently has bizarre dreams, and his game is mostly built on a weird bullet time gimmick (that is admittedly pretty great). In the last decade or so, Max went from hard-boiled city detective to the world's greatest alcoholic to Zach Galifianakis in The Hangover 2 cosplayer. Also one time he murdered all of the New Jersey mafia in a single night, which was weird.
If you've never seen the original Pink Panther or A Shot In The Dark (the best Pink Panther movie and the only one to not feature everyone's favorite wall insulation mascot), go see them, because Peter Sellers has one of the best comedic performances of all-time as the totally incompetent but baselessly confident detective. He solves cases, sure, but usually on accident and almost always with the help of others. Hell, he's even accomplished a seemingly impossible task - made a Steve Martin movie worse than Cheaper By the Dozen.
Let's not decorate this at all - Basil is literally Sherlock Holmes but a mouse. That's it. He even lives under Sherlock Holmes' home. The only things that are missing are the opium addiction and heavy sexual tension with Watson (named Dawson here). He even has a Moriarty-like nemesis in Professor Ratigan, who (in the Disney film, The Great Mouse Detective) attempts to replace the mouse-queen of England (wait they have mouse royalty?) with a robot. Knowing all of this should help explain why the BBC show Sherlock was initially advertised as "The Great Mouse Detective but with humans."