I have mice in my coop and rats in the run. I've had two different kind of chicken proof mouse traps in the coop, a live rat trap in the run, and a secured rat poison bait station under the coop with no luck even though I have been putting the feeders up every night. Last night I put some regular mouse snap traps under a small (8 or 10 inch I think) milk crate with a large rock (just in case the chickens messed with it before I got out there in the morning) in the coop. I had a pregnant mouse caught dead which is great so I'm trying it again tonight. I also borrowed a trail cam and put it down in my run last night; this morning I was able to see the routine of the rat(s) and I have a couple rat snap traps set up along their path. Fingers crossed for those. I am finding that regular snap traps are the best. You have to wait until everyone is put up to set them, and then pick them up before letting everyone out, so they are a pain, but they're worth it. I've read good things about electronic ones too, though they are harder to use in the run since they can't get wet. You could also try the bucket traps. If my snap traps start to fail I will give that a shot.
Really, mice and rats don't bother me, but I don't want them eating my eggs or chicks, chewing up my coop, or making tunnels for snakes to get into the run or coop. I know for a fact that the mice eat the eggs. The rats don't seem to care about them, oddly enough.