Kinsey Millhone is a private detective. Sue Grafton has written awonderful series about Kinsey, beginning with A is for Alibi andcontinuing, I believe, up to R. (Check the Book Series for some of thetitles.)
In C is for Corpse, Kinsey is hired by a young man who was in a terriblecar accident. In addition to the physical damage, he's suffered mentaldamage and memory loss. He is, nevertheless, left with the firmconviction that he was run off of the road and that someone tried tokill him. How Kinsey solves the case is very interesting, and this maybe one of the earlier books which reveals a great deal of Kinsey'spersonality.
All of these books are quick, fast-paced, compelling reads. For funmysteries, any of the alphabet series is a good bet.