One useful trick with both live and lethal traps is to leave them baited but unarmed for several days. Many have a switch or other gadget to let you do this.
This allows the little blighters to get used to them as a source of food, starting to make scent trails to the trap and treat it as a routine. Then you set them properly, and you'll get one animal after another.
Setting them straight away can just get the first one, while teaching the others to be wary. Once wary, you'll never catch them at all as they teach each other what to avoid.
I've tried a type of cage trap (for rats) which catches the first one with a spring-loaded door, then has another one-way door for its friends and relations -- you can get a whole family in there at once. In fact it didn't work all that well with rats, as they get very stressed indeed in the cage (even with food and shelter) and frighten the others off. It might be better with mice -- the mouse ones cost about a tenner, I think, from farm supplies shops.
Then there's the pepper trap. I doubt if it works, but it's so clever it ought to. You rest a brick on some stones to make a mouse-sized space underneath. You put your bait underneath, with some pepper on it. Mouse sniffs bait, sneezes, bangs head on brick, kaput!
Finally, don't just release live-caught animals "outside" -- they'll come straight back in. Take them a couple of miles away at least, preferably to the other side of a river.