SmallCaliberCentral

...I live in the city limits.

i, also live in the city limits. my suggestion is a pellet gun that will shoot at least 1000 fps. personally i like the 177 caliber, but you can find pellet guns in 22 caliber that are shooting over 1000 fps. choice of pellets does really make a difference. my rws model 34, 177 cal. likes the rws super dome pellets the best. in my experience with pellet guns, shooting over 1000 fps, most like a little heavier pellet better, and they are easier on the gun. the super dome pellets weigh 8.3 to 8.4 grains.
i have killed cats, birds, and various other vermin with mine at ranges from 25 up to around 50 yds. nice thing about pellet guns, is that they don't make a lot of noise.
one other suggestion would be to open a door or window, and stand aways back from it and shoot out of the house with a 22.
of course all of this depends on how close the neighbor are.
how close are these vermin, and how close are your neighbors? looks like a big yard or field, in your pics.

I have to agree with Futuretrades, an air rifle would fit the bill on this one! Gary

I normally shoot them with my bow. But I might let these few stay around for a while that way I will have some to shoot next year.

I'm part Indian but I don't think I could hit my foot with an arrow! We always leave some of our furry friends to "repopulate" for the next season. Gary

Well I think this would do the trick

My vote is for a bow, it's harder for them to get down their holes unless its a complete pass through.