You are not the first person to have this problem.
There exists a form of spring trap which comes in a plastic enclosure. They work by coming with bait inside and a hole in which the mouse enters to spring the trap, at which point the trap is just thrown away. Unless some one or something is small enough to get into that hole, they won't be able to get caught in the spring.
Or, just go to a Home Depot type store and just ask for children and pet safe traps. They should have several options available.
PS) Having a rat problem is very different than a mouse problem. Rats are a lot bigger and thus a lot more dangerous. A rat would probably just eat a typical mouse trap.