I live in the sticks and get rats here now and again. I hate killing things but do it when necessary. It's legal to use shotguns out here but rats are really hard to kill cleanly with one - they seem to squeal and kick about for ages afterwards and that's something I hate to be responsible for. By far the cleanest way to kill one is with an accurate head shot with an air rifle. I've found the most humane way is to set up my rifle upstairs and stick some birdseed to the patio with peanut butter and wait. They appear in regular time slots, early morning, mid morning, early afternoon, mid afternoon and then dusk. From above I can get a clean head shot and they're dead instantly. If you break up your silhouette with the curtains and a couple of tshirts dangling from the curtain rail they won't really see you if you keep still. I alway re-zero before lying in wait - you'd be amazed how much different your point of impact is when you change distances so for a clean kill you need to be pinpoint accurate. I use a .22 BSA Scorpion T10 but at such short range almost anything firing at 10 ft/lbs or more will do the trick instantly. Afterwards I pick them up on a shovel (if injured they'll bite you if they get the chance and you may not realise they're not dead yet) and bung them into the household wheelie bin. If there's blood on the patio I wash that down with a mild bleach solution just in case then rinse it off with a bucket of water so the dogs don't get anything irritating on their paws.

As far as rifles go, I really like the killing power of the .22 at close range and the flat trajectory of .177 at longer ranges so I have one of each in position and I'm ready to go as soon as a rat is seen either at the top or the bottom of the garden. I've found (personal opinion) that springers seem more accurate if they're in .177 than .22 I reckon it's the movement of the internals having to work harder to push the larger pellet but the rifle just doesn't seem to move as much in the smaller calibre - so that's worth investigating for yourself if you want to reduce wounding as much as possible. I agree with the people who rate a secondhand Weihrauch for the job - I have an HW80 that's thoroughly reliable and if I had to use a springer then that'd be the one I'd use.

BTW - if anyone tells you that all Jack Russells are excellent ratters then that isn't so - my two will kill a rabbit or a pheasant in a heartbeat but won't go near a rat. Maybe it's the smell or maybe a sixth sense but they absolutely refuse to be interested in rats.

Good luck - and remember, killing a rat with a rifle (assuming you can shoot straight) is more humane than poisoning them or drowning them, so if like me you hate killing things then it's worth remembering that if you don't do it with a clean and accurate shot then someone else will do it with poison, a badly placed shot or worse.