Good thread... I'm late to it... but many good replies of "to each their own" takes on what pests are to them and elimination of said pests.
My backyard doesn't ever see rats, I know there are mice around but I never see them or seek them out. Chippers on the other hand, get no free pass and my backyard must smell like death ... 'cause I've not seen one to shoot in a few weeks, their digging / tunneling brought me to this point several years ago.... I've kept a tally and it's close to 100, and that's only confirmed body count,,, made solid hits on others and they made it to a hole, likely becoming worm food is okay but not tallied.
I have lots of fox squirrels and they get a pass, they're non-destructive unless you figure the twigs they cut for nesting and allow to fall into the yard or the buried acorns they forget about and a new oak sapling shoots up... only to succumb to the mower... non-destructive and fun to watch their antics. Now red squirrels are another situation, I've dealt with over my years at other properties - mine and in-laws. Reds are destructive and WILL chew their way into a building if they choose to gain access... a good red is a dead red!!
Invasive birds.... I'll occaisionally shoot house sparrows when I witness them bullying the house wrens for territory. Starlings and cowbirds also get no pass in my backyard. Crows are nest thieves, but also do good cleaning up roadkill and are protected by law during their nesting season here in Ohio.
I've never shot a cat, but they aren't a favorite of mine either, mostly 'cause other neighbors have cats and "at times" will allow the "house cats" out to roam and these litter box trained house cats like to use my decorative flower beds / mulch for their own outdoor litter box.
Raccoons / trash pandas,, ALL succumb to "lead poisoning" if they're witnessed in my backyard and a scope reticle can be put on them! .. nuff said... buzzards gotta eat too. ;-)