Truthfully, if going with a backyard thing, a .22 in a slower-than-sound pellet will give you best varmint results, with a corresponding lack of neighbor issues.
This doesn't mean that a .177 is bad---but if the pellet isn't going that fast, a heavier larger pellet is going to help.
If supersonic noise isn't an issue, then you can buy some RIDICULOUSLY overpowered spring guns out there in .177.
(I have always wondered about the "sound suppression" devices, though. There isn't any noise from gas or burning powder, so what exactly is being suppressed? The spring or piston? How does hanging something off the front end change that? Does the compressed air, driven by the piston, make THAT much noise that baffles in the front will actually make any sort of difference? It certainly won't change the supersonic crack is the pellet is going fast enough. What is it for?)