I tried to go the humane route-bought a big cage trap-it's not really humane at all though. You are supposed to check the trap regularly (every hour I think!) well I set it up at night since that's when they are most frequent and who would get up regularly during the night, go all the way down the garden in the pitch black to check? And then what to do with it if you do find one in the middle of the night? You still have to kill it somehow.
Rats usually survive but if a mouse is caught I will most often find it dead (and rats are pretty smart when it comes to the cage traps-I have caught mainly mice and only a few smaller rats). Not only that but if you do manage to catch a live rat (which will have been stressed out in the cage for a while) and you release it in the country thinking you are being kind, apparently it is pretty sure to die anyway without it's usual food and water sources. So all that being said, me being an animal loving vegetarian who tried to go the humane way-might have caused an even bigger problem and more rats and may have to resort to poison etc. in the end anyway-killing more rats and causing more suffering than someone who just kills them right away. Sigh, keeping chickens is not easy!
I would say-make sure there is no way they can get to any food (mine went away mostly when i put all food out of their reach and bought a metal container only the birds could operate-they came back when i had ducklings and food out for them a few months ago) Block off all their holes, and if that fails you may have to resort to killing them (not like i have to convince many people on here anyway!) I am still hoping I don't have to resort to poison, but very aware of the problem escalating.