Have you ever seen Murder By Death? It’s a movie that does pretty much what you’re thinking, except it stars more than one person.
Gregory McDonald’s Francis Xavier Flynn
Remington Steele!
David Addison
And, of course, Dirk Gently 
Here’s a vote for Ellery Queen.
Oh yes. I especially love Wexford–I’ve been rereading a lot of those mysteries lately.
Dr. Augustus S.F.X. Van Dusen, the Thinking Machine
Mr. Linley (“Solely in order to get an appetite.”)
(Both get their deserved fame for a single story).
The Hardy Boys
Nancy Drew
Plus, you wouldn’t really need a costume to portray him – all you’d have to do is talk while your head is cocked at a 90-degree angle.
Napoleon Bonaparte - the half aboriginal detective by Arthur Upfield.
The tortured Lorraine Page from Lynda La Plante’s ‘Cold Shoulder’ series.
How about the earliest one of them all, Brother Cadfael?
I gotta echo Dirk Gently, but my favorite crime stoppers have to be, and I know this is not a detective, Perry Mason, Mason could always find some obscure to pin the rap on. The other one is Detective Nose. If you can solve a mystery in two minutes or less you have to be good.
SSG Schwartz
I think part of the trick here is finding recognizable characters. Given that, Columbo would be easy to accomplish, although I don’t know if Brits know him as well as Americans.
I can’t believe I get to be the first to say Fox Mulder and Dana Scully of the X-Files, my two favorites. I also loved Veronica Mars, Lenny Briscoe (I miss Jerry Orbach-- I can’t watch Law & Order without him), Jimmy McNulty and Lester Freamon (from The Wire).
Surely someone mentioned Sam Spade and I just missed it. :dubious:
Yep, see Post #3
Darn! You beat me! Great character & awesome detective. If I’d killed someone & then he showed up to investigate the death, I’d just confess & save myself some angst.
First thing I thought of when I read the OP. Great movie, one of my all-time favorites.
For your purposes, you need detectives with styles, mannerisms, tics, quirks, habits, accents and catch phrases that are readily recognizable and easy to imitate or convey. A LOT of great fictional detectives (Adam Dalgliesh, for instance) would be terrible choices for you, because he doesn’t have any of those things. You could do a wonder performance as Dalgliesh, but most of the audience wouldn’t even grasp who you were playing.
The sketch probably needs to be broad, and should focus on “types.” The mix should include
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Sherlock Holmes goes without saying
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A tough guy, urban American private eye. Sam Spade works (you can just do Bogart all the way), but so do Mike Hammer and Philip Marlowe.
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Sherlock Holmes, obviously.
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A recognizable TV character (Columbo comes to mind, but McCloud could be fun)
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A detective from a kids’ series- Nancy Drew or Encyclopedia Brown, perhaps?
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A “blaxploitation” detective like Shaft.
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Nero Wolfe would give you the opportunity to guzzle beer, say “Pfui” and make fat jokes.
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A cartoon detective like Dick Tracy!
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Some ethnic detectives with funny foreign accents- try Hercule Poirot and Charlie Chan.
This is the movie that made me think of Charlie Chan, thanks to Peter Sellers’ performance.
“No pulse, no heartbeat. If condition does not change, this man is dead.”
The wife’s and my favorite: Lieutenant Joe Leaphorn of the Navajo Tribal Police in Tony Hillerman’s excellent series. (With Sergeant Jim Chee of the same force a close second.)
This guy is much earlier.