Do you have neighbors? How close are they? Would you consider yourself as living out in the country? I ask because you could live on the edge of a town and still have these animals around.
It sounds like you would enjoy shooting, if you don't have any neighbors for a good long distance, then the .22 is the way to go. If you do have neighbors but with pretty good spacing between lots(similar to a city block), something like a high quality Beeman air rifle(using lead pellets) will do the job as well. Keeping shot placement in mind and as always being aware of what is beyond the target(slugs, pellets, and BB's can ricochet and/or skip along the ground for some distance before finding a place to rest).
The down side to traps is that you have to take the little diggers out of the trap, exposing you to possibly disease carrying fleas. Often times trapped diggers are still alive when you discover them, which may or may not be a problem for you. If you have pets, traps can be a risk depending on the style of trap. Sometimes traps are the only option.
If the holes are within distance of a garden hose, then get yourself a shovel in hand, put the hose in the hole and flood'em out. Ever play whack-a-mole?
Poison? Never.
To me, shooting when possible is the way to go. Find a good place to sit and shoot for awhile(if your lucky that might just be on the patio under some shade), then go out with the shovel and push them into the hole a bit, if there is a mound then you use that to fill the entrance to the hole. A digging fork also works well to colapse the entrance to the hole, and bury the digger..now fertilizer. I could also make comments about the hours of entertainment, but I'm sure that would offend someone ;)