Pictured on your left is one of our Puggles, whose name is Chauncey Gardner. I suspect that pound-for-pound this 20-pound dog is capable of excavating more cubic feet of earth than just about any other dog.
I am not sure what sort of critter Chauncey thinks he is going to catch with his digging. I have been filling in the 3-foot hole every day, but in 30 minutes or so Chauncey is able to revert the ground to its former holey state. I suppose I could just let him dig in this same spot: it is out of the way, it is not near the root systems of any significant plants, and the time he spends on this massive hole he is not digging other holes in my yard.
Left: Momentarily surprised Puggle
I had to set up the second photo, and I was shooting for a Scooby-Doo "ruh-roh" feel in the next image. I called Chauncey's name in an excited voice and he popped up to see what was going on. I managed to get lucky and snap the shutter at the right moment, and the picture came out pretty much as I envisioned it.
It is difficult to get angry at such a happy little dog, and his digging is just a minor quirk as far as I am concerned. Heck, for all I know he might be inadvertently aerating the earth in the places he digs.