I used to read mystery novels when I was a preteen but I have not read very many mystery novels in the last few years. I sort of got bored of them, to many were either too easy to figure out (the author gave too many clues) or impossible to solve (lack of clues or erroneous, not misleading - misleading are good, clues). I have considered writing a mystery novel, I have even attempted to back when I was very interested in them, and I have considered writing one during my current brainstorming session.
Mystery is not a genre that many people choose to write Ranma fan fics for; there are a few though. The main reason is probably because it is easier to write Ranma fan fics in other genres. There are ways to do so, for example, Ranma disappears and someone eventually calls the police and they suspect murder. There are plenty of suspects. It is not unbelievable considering Ryoga had planned to kill Ranma and bury his body in the woods (even went so far as to dig the grave). Contrary to some fanon police do exist in the Ranmaverse both in the manga and the anime (for example, Ranma jokes about calling the police for sibling abuse when he pretended to be Ryoga's sister, during the Orachi arc one of the characters dresses like a policewoman, and Ryoga has asked at a police box for directions).
Though I'd more likely choose to do a crossover than a pure Ranma fan fic. Possibly Detective Conan. For an example of how to start the story: A murder has been committed and a rich man has gathered together several private detectives including Shinichi Kudo (who for obvious reasons can't show up), Kogorou Mouri, and Heiji Hattori. In addition to them they also asked the Tendo dojo for help (they are asked/called up to help people several times e.g. the Chonaikai have done so a few times in the manga, a mayor of a town being besieged by an octopus jar came to them for help, they got a phone call to deal with monster sightings, etc.).
This would be Ranma's first time dealing with a murder mystery, he is usually called upon to deal with the supernatural (Oni, Horse kami, living octopus jar, etc.), though he is sometimes asked to figure out who is committing a crime (such as figure out who is attacking people with pig and pony-tails or to stop a panty thief). Ranma does have many of the skills required, though not always from the side of law (for example Genma has shown the ability to break into safes and Ranma has stolen things). He has searched several peoples rooms, is good at finding out people's weakness and secrets (he tends to use them against people, though this is more commonly attributed to Nabiki who generally just takes advantage of a situation in the manga), is usually the first person to figure things out (like that Tatewaki was the principal's son), he is very observant when he wants to be (can figure out techniques by having seen them only once), he is naturally suspicious of people (for example, Nabiki a few times or Happosai's friend lucky when he was pretending to just be a child), etc.
There are also several things that could be done to make the two universes and the story flow better. One thing would be to make those books we see Ranma reading fairly often, but most of the time can not tell as to what they are, mystery novels and manga. Another thing would be to have Ranma had been in a mystery club in junior high school (we know very little of Ranma's past and since Ranma hates telling people anything about himself if he can avoid it, it is likely that even the other casts members would not know). Yet another thing would be to change Ranma's physical abilities from blatantly super human to borderline super human. The super natural things would not have to be excised from the story since Detective Conan has things that do not fit with our reality as well. Things like the poison given to make Shinichi young (which could be explained as having one of its ingredients be a mushroom of aging), the perfect disguises of kaitou kid, that detectives and police workers allow a ten year old around a crime scene, none of the side effects from Conan constantly using tranquilizers on Kogorou Mouri (which are particularly dangerous in combination with other depressants, such as alcohol which he drinks a lot of), that the creator of Sherlock Holmes (Sir Arthur Conan Doyle) believed in fairy creatures enough to be tricked by a couple of teenage girls using trick photography to make fake fairy creatures, the large number of murders that happen near him (which is lampshaded in the series), the FBI having jurisdiction in Japan (though I do understand why the mangaka chose to use the FBI since Japan does not have an organization comparable to the FBI - Japan has the NPA and Naicho but they are considerably different from the FBI), Several characters like Ran's and Kazuha believe in ghosts, Conan has premonitions, Ai Haibara can sense the Black Organization members, etc.