While rats and mice are both pests in the coop, they do have some differences. Rats can travel a fairly far distance, so nests may be far away. They particularly like to be under buildings and in brush and debris piles. Mice are usually a bit closer. I don't like poison since it doesn't tend to stay where you put it. They eat it, move on, and possibly poison another animal that eats them, possibly including a chicken. Mice can even squeeze through 1/2" hardware cloth, I had one get fat and get stuck in the wire. My chickens will eat a mouse if they see it, but I find they tend to run in the coop more after dark when the chickens are sleeping, and they can't really see them then. The mice (and rats) are after food, so keeping it cleaned up is important, but that is really difficult since chickens tend to throw it around everywhere when digging in the feeders. It's almost impossible to not have ANY feed on the ground somewhere. I find the mice will also go after undigested grains in the droppings on the poop board at night, so another source for them. I've tried just about every trap out there, and for many reasons have settled on the electric ones. They are as humane as you can get, last a long time since they are reusable, clean up is easy, and they are safe for the birds. I put them in the coop around the edges where the mice tend to run. In the fall and winter I sometimes get at least one a night, sometimes several. I leave them in the coop year round, but in the spring and summer I don't catch nearly as often. I hate sticky traps because I don't want to deal with what to do with a live mouse, and I don't like having to try to remove other things from them that inadvertently get caught, like snakes and lizards. Particularly since the snakes help keep the mouse population down. Bucket traps work, just not my thing, I find the electric ones easier and more convenient, but yes they do cost more.
https://www.raising-happy-chickens.com/rats-and-mice.html
https://www.raising-happy-chickens.com/rats-and-mice.html