We messed around trying to catch and trap a rat for a few weeks with no success! My husband is very creative and determined and he tried several methods and traps. Once you have one rat you'll soon have a real problem, so we reluctantly looked into poison. Our local feed store recommended "Just One Bite" rodent poison in the hole we found. That took care of it. Unfortunately, the EPA made new rules so you have to buy it in an 8 lb. box, so there's enough there to take care of thousands and thousands of rodents.
I agree. One rat seen = an infestation already in place. You can try trapping, shooting, drowning, but that is not likely to clear the problem.
With poison, be sure to place it in such a way that no other animals can get to it.
Also, remove the food source: keep all feed in metal cans with lids. Never leave feed out overnight. Is your coop rat proof? Rats can wreak havoc on young birds, nests with eggs. If your flock spills feed, that right there is a source of rat food. Got bird feeders out for the wild birds? More rat food. Finally, consider switching to fermented feed. This will clear up the source of chicken feed for the rats. Generally, fermented feed is all gone by evening. It's not likely to be spilled, and even if it is, it's not easily transported back to the rat's nest. Fermented feed will also give your flock superior nutrition while cutting your feed bill. It's a win/win/win proposition.