When we lived in South Africa, we had a minor problem with mice. We could hear them scurrying around in the ceiling after we went to bed. My enterprising father put a trap in the ceiling next to the attic door.
Well we woke up one night from a dead sleep around 0100 to such a racket I thought The Spear of the Nation was coming through the roof. An significant about of clomping and whoming sounds. Come to find out it was just a good sized field mouse experiencing his death throes. I think we used peanut butter as the bait.
I also support the notion of outdoor cats. Not being a cat person myself I think all cats should work for a living.