According to Sean Coyne in The Story Grid, the core value in a story set within a crime genre is justice vs. injustice with the core event being the exposure of a criminal. The inciting incident is generally some unjust occurrence that throws the protagonist out of his or her comfort zone and on a path to restore justice. There are several different categories within the crime genre.
The most famous is the murder mystery. It is in fact so predominant that it gets subgenres of its very own.The inciting incident is nearly always a dead body and the story often concludes with the reveal of the murderer. (If the murderer is known to the audience it moves more into thriller category than mystery). Conventions of the genre include red herrings, interviews with characters who have their own secrets and agendas, a slow reveal of clues, and a demonstration at the end of how the clues fit together. Within the murder mystery are even more sub genres.
The Master Detective- Think Sherlock Holmes. It’s pretty much the trope setter. Well… it might as well be.
Cozy Mystery- A non-detective with skills in seemingly unrelated areas (like writing novels, for instance) finds their skills and experiences make them surprisingly and uniquely qualified to solve the case. A good example of this is Aphrodite where I more than dabbled with the cozy mystery sub genre within my paranormal romance.
The Cat Mystery- Cats solve crimes. Enough said. Diane Duane has a great series set in the same universe as the Young Wizards Series that pulls in this sub genre perfectly in The Book of Night with Moon and To Visit the Queen.
Historical Murder Mystery- A mystery set in a historical time period or featuring a historical figure. But Kaitlin, you might be saying, wouldn’t it have to be set in a historical period if it featured a historical figure? To which I say Sleepy Hollow.
Noir– Noir is as much a style as it is a genre. It features hardboiled detectives and/or lawyers and/or vigilantes, lots of dark backgrounds (though the reverse has been done successfully), femme fatales. It’s often told in flashback “(There I was, sitting in my office, when a dame walked in. She was trouble.”
I actually took a class on Noir Fiction in College for my Topics in American Literature elective, it was fun. My favorite was the one about a guy who goes to a police station to report a murder. Who’s the victim, they asked. “Me,” he replied. He’d been poisoned and the rest of the movie was him telling them who-done-it. Batman is stylized after Noir mysteries, and a lot of popular TV shows have done at least an episode in the Noir style.
Paranormal- This is really more a cross-genre between paranormal (often romance) fiction and crime fiction. It can crossover with any of the above categories and magic users of some kind. The Hollows Series by Kim Harrison is one example of just straight paranormal romance mixed with crime fiction. There’s also a lot of historical fantasy crime fiction that sets magical people back in time solving mysteries (to some degree, the His Infernal Devices fits into this). Paranormal pairs well with everything.
Police Procedural- This is your Law and Order/CSI/Dexter type stuff. This one also pairs well with paranormal.