Kinsey Millhone is a private detective. Sue Grafton has written a wonderful series about Kinsey, beginning with A is for Alibi and continuing, I believe, up to R. (Check the Book Series for some of the titles.)
In C is for Corpse, Kinsey is hired by a young man who was in a terrible car accident. In addition to the physical damage, he's suffered mental damage and memory loss. He is, nevertheless, left with the firm conviction that he was run off of the road and that someone tried to kill him. How Kinsey solves the case is very interesting, and this may be one of the earlier books which reveals a great deal of Kinsey's personality.
All of these books are quick, fast-paced, compelling reads. For fun mysteries, any of the alphabet series is a good bet.