I just became thane in Whiterun and was wondering, What crimes can you get away with as thane, and how many times can you use your rank as an excuse for crime?
The thane title can allow you to get away with a murder once and numerous smaller crimes. If your bounty is really low, like from a single lockpicking, just say "Do you really have time for this?" No reason to use the Thane option. Also:
guards will forgive your bounty up to 1999 gold (once per hold).