Well I've never spent more than about 30 minutes playing whack-a-mole with rats, and usually it's between 5 and 10 minutes, doing a quiet perimeter check or two. If I don't hear any high-pitched communication between rats within the first minute outside, I know it's likely to be a failed mission. Rats talk to each other a lot. Of course if I'm bashing around they'll go quiet and I won't know they're around, but I'm quiet, and they keep up the intermittent squealing a few times per minute if I don't spook them. Once I've shot a rat I won't hear any of this for at least 15 minutes. They hear the thrashing of hind legs and go very quiet, knowing one of their comrades has fallen.
I see live rats in the daytime at least a couple of times per week, usually from my workshop window, where I can see much of my neighbour's garden and back porch. Rats love her place for the bugs in the garden and the smell of various pickling/fermenting foods on her porch. They never get into the jars/crocks, but they're attracted to the smells anyway and hang around. So about half the rats I shoot are on her porch floor, usually from the top of my fence, at a range of 5 yards. Holdover becomes important when so close to target, scope-over-bore height making for rapid changes between 3 and 10 yards. I've memorized it thoroughly so I always place the pellet into the base of the brain.
As for other methods - as I said, cats are next to useless, except for one who sometimes manages to get a rat but is usually just a little too slow - he's more than 10 years old and a bit fat. And even he won't do more than poke at a squirrel. They're all interested in chasing squirrels, but if they catch one they immediately let it go, discovering that squirrels bite back. Most will just get a foot or two away, see the squirrel squaring off and yelling at them, then turn and hide. When I drop a squirrel there's usually a cat around that's been watching it, and that cat will run in right away to check it out... then paw at the spasming squirrel a few times, maybe even drag it for a few seconds, then walk away, no longer interested. I chuck them in the city compost bin, layered with leaves and weeds. Makes for higher quality compost on city parks. If SHTF I'll start eating them instead.