My main inspiration for asking this question is the events around Jack the Ripper, history's first celebrated serial killer. Was there ever much chance of catching him/her or was it just that mistakes were made early on? Did murderers get away with their crimes a lot more easily before fingerprints and DNA were used?
Topic Overview with several good threads:
Crime and punishment in the past
One comment related specifically to Jack the Ripper:
http://www.reddit.com/r/AskHistorians/comments/sipwy/how_were_crimes_investigated_in_the_past/c4eoeus
I watched a documentary on Jack the Ripper and one of the policeman remarked that whenever a crime is committed their first course of action is to quickly begin pounding on door and questioning everybody in the vicinity if they saw something.