I've seen people mention in various places that there are no police in the Ranmaverse. In my opinion there is a police force in the Ranmaverse whether you go by manga or anime canon. We see Ryoga stop at a police box and ask for directions to the Tendo dojo, the forgetful boy Shinnosuke dresses as a police woman, and Ranma comments that he should call the police on Ryoga for sibling abuse after Ranma pretended to be Ryoga's sister Yoiko are just a few times they are mentioned.
Following is speculation:
Depending on the nature of the crime/problem the police may not be the ones you'd call. It is demonstrated repeatedly that the Tendo Dojo gets called to deal with monsters/supernatural things. If you live in the Ranmaverse and have a supernatural problem you go to a martial artist or a priest not to a police officer.
Why don't the police come and arrest Ranma and company for the various crimes committed? Not really that hard to explain. First lets deal with the more serious crimes: murder attempts and kidnappings in the manga. The reason the police don't get involved in those is fairly simple, they are never reported. Ranma and the other martial artists rather deal with those things themselves and would not report them to the police. Also the Japanese have a different view on police than the west. The West (USA in particular) has many lawyers (more than a million in the USA), the Japanese have less than 50,000. The Japanese do not like going to the police except as a last resort.
Many crimes have also passed the statute of limitations. Genma can't be arrested for stealing Ukyo's cart because too much time has passed.
Why hasn't Happosai been arrested? Can you see any police force consisting of regular people in the world being able to catch Happosai if he didn't want to be caught or managing to keep him imprisoned if they did catch him (possibly by using female cops). Besides that point, Panty theft is not exactly high priority and there have been attempts to stop him. An association in Japan that would be similar to a US Neighborhood watch goes and asks the Tendos to stop the panty thief as well as the niku-men (bald men who wanted the dragon whisker) in the manga. The people with the responsibility for stopping Happosai are also the people he is living with, the vast majority of the people in the Ranmaverse probably don't know that Happosai lives there (the Tendo's and Saotome's would deny having anything to do with him).
Property damage why aren't they arrested for that. Like the other things this isn't all that hard to explain. First off the damage is often exaggerated in fan fiction. The most damaging battles were done out in the wilderness (breaking point fight, Herb battle, Saffron battle, etc.) or had no witnesses (Ryu battle). The next most common battles are prearranged with seating for bystandards even set up - martial arts dining, cheerleading, ice skating, Ukyo's battle versus Ranma, rhythmic gymnastics, ki blast battle, etc., in which some damage would be expected. Even in those cases the damage is usually confined take the ki blast battle between Ranma and Ryoga, for example, the damage done was confined to a soccer field. After that the most damage occurs at the Tendo's house followed by the school. Tendo's wouldn't report it as for the school who knows Principal Kuno may just not want the cops looking in on how he's running the place.
As for Ranma being arrested for property damage remember most of the damage caused by Ranma is usually done in self defense or from someone slamming him through a wall by people who don't stick around (like Ryoga, Taro, etc.). Many of the people who might be arrested don't stick around, they leave the area (and some the country - like Taro). Another reason it wouldn't be reported is quite simple, abilities like Ranma's are not very common we are shown this several times in the manga, so who would believe you if you did report it if they didn't see it themselves. Imagine trying to report these: "Hello police a girl just shy of being 5 foot tall just punched my wall down", or "Hello I'd like to report a battle between a Hindu goddess and a huge flying monster with tentacles. No this is not a prank. No I'm not on drugs. Of course I'm not insane a Hindu goddess did strike my house with lightning".
Even if the police did come to investigate they don't have any proof (punching a wall to pieces doesn't leave much evidence behind as to who did it) and lets say they did send someone to the Tendo dojo. Does anyone there look like they are capable of tearing a building down with their bare hands? Imagine the scene. Police come to arrest Ranma and all they see is a cute little girl who looks completely defenseless and like she's about to cry (Ranma would be acting and has used this type of acting before). Also several cast members are fully capable of altering the memories of anyone sent to investigate. Happosai, Cologne, Shampoo, Ranma (he's used suggestion incense twice in the manga and may know more tricks), and possibly Genma (might or might not but considering he can disappear or turn into a Panda good luck finding him) and Kodachi (one of her drugs/poisons could probably do that) can all alter people's memories so they believe that someone else did it or they have the wrong people or something. The Kuno's appear to be rich enough to pay the police and such to look away from their offenses.
Many of the things ascribed to be legal matters in fan fiction are not. For example the various engagements. Those are not legal matters they are matters of honor. Legally nothing can really be done about them. As far as the law is concerned they don't exist since in Japan it is illegal to force someone to marry (happens anyways though).