I don’t like the glue traps because they don’t kill the mouse, just get the mouse good and stuck. You then have a live mouse to dispose of. Sometimes they will try to get free by chewing on themselves. I know glue traps work, but I’ve never been able to bring myself to use them.
Regular mouse traps worked best for us. They are quick, and I’ve never had a mouse suffer in one. Once I tried some other kind of trap that didn’t use bait. You put it along an area where the mouse typically runs, and when the mouse goes through, it clamps down. Once I was horrified to find the trap had closed on the back half of the mouse’s legs. It was alive, and had dragged the trap halfway across the house. I felt so bad, and never used those again. Stick with the old-fashioned traps and a bit of peanut butter.
The catch-and-release traps work really well too, but you have to release the mouse far from your home or it will just come back. You can’t plop it into the back yard.
Where there is one mouse, there are usually more. They breed quickly and, besides the health hazard of droppings and fleas, they can do quite a bit of damage. We had a mouse problem once and were using the catch-and-release traps. There were too many mice though (we’d just bought this house not knowing the mice were included). One day, I went down to the basement and found a big puddle in the laundry room. The mice had chewed through the drain hose from the dishwasher, so whenever we ran the dishwasher water poured onto the basement floor. After that, we got a kind of poison that makes the mice very thirsty before they die. They go outside to find water and expire. It worked—no more mice, and we never had any hint of a mouse corpse somewhere in the walls.
As for cats, they aren’t always a deterrent for mice. I grew up in the country, and our multiple cats didn’t seem to scare the mice. Most of our cats weren’t “mousers” and typically only caught mice that had already been caught in a mousetrap. My inlaws got a cat to combat their mouse problem. One day my father in law saw a mouse in the bathroom. He put the can in the bathroom and shut the door. A while later, he peeked into the bathroom and saw the cat and mouse just staring at each other. After that he bought some traps.