Jack Russell
Use a proper break-it's-neck type trap. I had to modify mine as they were very good at taking the food without setting it off - I simply screwed the food down to the bait plate. Biscuits worked the best, crunch creams being a particular favourite of the rats (and mine).They don't really make a mess, a small trickle of blood maybe. Be prepared though - they can end up trapped by a leg or tail and still be alive. Have a suitable length of wood ready to twat it round the head and finish it right off.
+1 for proper traps ....With mice I found smooth peanut butter worked well as bait - they can't pull a lump off to eat somewhere else so have to put their head in the trap to lick the bait and are killed instantly when it goes off.