The first story if Isaac Asimov's "The Singing Bells" in which valuable moon artefacts are stolen. The baffled cops seek out the Great Detective who can solve the case without leaving his home. The next two stories, Poul Anderson's "The Martian Crown Jewels" and Philip Jose Farmer's "A Scarletin Study" are Sherlock Holmes pastiches. In Anderson's story, the Holmesian detective is a Martian creature who looks more like a bird than like Basil Rathbone. In Farmer's tale, the detective is a talking dog.
Then comes Donald Westlake's "The Winner," which is a prison break story. No detectives in sight.
Finally, with Tom Reamy's "The Detweiler Boy," we get what the picture and title promised, minus the fedora and trenchcoat. Reamy's tale is even set in Los Angeles, though his first-person narrator finds a killer who's nothing like anyone Philip Marlowe ever tracked down.
In "Time Exposures" by Wilson Tucker we have cops. No private-eyes to be seen. I got a kick out of this one because while the main character has a camera that can photograph the past, he has to use a slide rule to figure out the settings.
Robert Silverberg gives us a continent-spanning city in "Getting Across," and while a private citizen is asked to find a missing woman, it's a real stretch to call this one a private-eye story.
And there's just no way at all to call Philip K. Dick's "War Games" even a cop story. No cops, no private-eyes, just people who evaluate toys.
The last story is one of Larry Niven's Gil Hamilton stories. The same two editors chose to close out Supernatural Sleuths with a Gil Hamilton story, as well (my review of that book is here). As I mention in that review, Hamilton's not really a p. i. He works for a branch of the U. N. called A.R.M., and he hunts organleggers. But he's close enough to a p. i. to qualify, especially in this company.
Now don't get me wrong. I'm not saying the stories aren't good ones. In fact, I liked them all. Niven's is the best of the lot, for me, but they all have something to recommend them. If you're looking for cross-genre crime stories, this book is the place to find some entertaining ones.