One useful skill for a psychic detective is psychometry, the art of picking up information from holding an object, or from visiting a place. Psychics skilled in this art can pick up information about a person from holding an object belonging to them, or visit the scene of the crime and receive impressions of what happened there. This is very useful for locating missing people, tracking criminals or finding evidence.
There are some famous cases where psychics have helped the police, for instance a psychic medium called Robert James Lee was believed to have helped the police in the case of Jack the Ripper by receiving information from the dead victims.
Unfortunately many are frauds, sometimes involving other people to help the so-called psychic so they look like they have psychic powers but don't. Some psychic detectives use a technique called shotgunning: giving a large amount of information, some of which is bound to be appropriate. Usually this works because the person remembers the information that was correct, and forgets the information that wasn't.
There were message boards for posting the whereabouts of Osama Bin Laden so psychics could share information they received using ESP, the ability to receive information not perceived through the usual five senses. This can be in the form of clairvoyance, which is receiving psychic information as images, clairaudience, which is hearing information, or clairsentience, which is feeling things outside the usual range of senses.
However it is hard to prove or disprove instances of psychics helping to solve crimes using their psychic abilities due to lack of evidence. For instance there are psychics who claim to have foreseen J F Kennedy's assassination but this is very hard to prove after the event. This is particularly true when assessing early cases of psychics helping the police because there is not enough information to assess whether they were genuine or not.