Jack Rusell dogs are good for getting mice and rats. That's what they were developed for in Europe, and do it instinctivly. Certain other dogs can do it well, also, but I know my neighbor's Jack killed at least one mouse in their apartment. They'll even go after squirrels in the street, so it's important to keep them leashed.

Most cats have to be taught as kittens, by their mother, how to kill rodents. A weed dealer that I used to know had house cats that were ineffective at killing mice, and would just play with them It was funny, this guy would yell at the cat, "THIS is what YOU'RE supposed to do!", while hitting the mouse with an object.