Here's what I do:
I lock the chickens in the coop and put snap traps IN the enclosed run where the pesky rats go AT NIGHT. this keeps other innocent creatures away for the most part, as my run is pretty secure with 1/2" hardware cloth and a metal roof, but without concrete on the bottom, sneaky rats/mice can get in. Any critter in the run at night is presumed to be GUILTY! Just be very careful that the chickens can't get to the snap traps! Then in the morning I remove the traps and let the girls out.
If you don't have a lockable coop but the coop/run combo is lockable, then you could put the traps outside the coop at night. The challenge for me is to get the traps removed before other innocent critters wake up - birds for example wake up darn early. I killed a bird in a snap trap and felt terrible. That's why I now put it inside the run with the girls locked in the coop. Squirrels don't come out at night.
I also remove the food EVERY night and I lock it up inside my garage (no rats have gotten in thus far). This is very important. Otherwise you are feeding rats and their babies and their babies and their babies...ugh. Also, I try to remove the food a bit before sunset as my girls are messy eaters and spread food on the ground - this way, they clean up the ground a bit before going to bed.
I'm aware of the anti-pet perspective and wouldn't try to debate it (because my dogs & I know who's right
), but perhaps a rescue spayed/neutered feral cat could be of help here. They wouldn't be doing anything harmful (like killing birds) that they wouldn't otherwise be doing somewhere else, and you'd get the benefit of rat control. I have been considering this myself.
I've also been experimenting with a plastic owl that has a motion sensor on it and it hoots if it detects motion (note: the chickens are DEATHLY afraid of it and I only turn it on at night), and with the rat zapper traps. No opinion on them yet.
I do not use poison. I am too worried the chickens will get to the poison, or to a poisoned rat, or that another animal will get to it. I was a vet tech at one time and have seen too many dogs and cats die a horrible death from rat poison.