Cats dont work,some are right lazy bleeders and couldnt catch a cold,others actually like to bring mice in to play with so if the shock doesnt kill it you actual end up with a mouse in your house that was previously rodent free.
Breakback(spring)traps are very effective if set up right,they must be along the wall edges with the trap at right angles to the wall(a lot of people set them up inline which is not as effective.
Sticky chocolate(like Mars) is effective as anything.
As said previously you need to seal up any ingress points and gaps to prevent them free movement.
Wirewool,or Wall filler laced with Broken/Ground glass will prevent them gnawing back thru.
If you can get hold of them then “Sticky Boards” are very effective ,they are a piece of card with VERY strong non setting glue on them,place them along the edge of walls like a normal trap ,the mouse cannot see them and once on it will not be coming off it again.
Ex-pest control engineer so I know the above will work.