Question
We had an aspen in our backyard that took major storm damage and had to be removed a couple of years ago. The company that removed it ground the stump, but left several of the large roots and they have been sprouting new trees. We've been cutting them, but to perfectly honest, I'd love to have another tree in that general area, because it provided a great deal of shade to our yard. Is it possible to let one of the suckers grow into a new tree, or would they be too unstable?

Answer
If they have their own root system they can be dug and transplanted. Check the sprouts root system it  maybe from the parent root (large root) and if so it will be difficult to dug a good root system to transplant. It would be worth the try but the success will be better if you wait until he hardwood drop their leaves in the fall. Transplanting trees in the hot summer give them a lower chance to survive.