I've found that when patrolling the town looking for potential troublemakers and wanted men, Police will be doing a similar job as normal, when something happens I'll hang around the scene, because often the Police are outnumbered significantly and if anything kicks off I can be on hand to assist or take down potential wanted men to jail.
I've also found that if there's a lot going on at the scene, the police will hand a confirmed criminal over to myself to be transported to Prison so they can get on with dealing with the rest of the hobo's.