Don't cut them off, get right down into the dirt and rip them off at the base. I just went to a pruning clinic today and that was the nursery owner's suggestion. You have to get them right back to the trunk of the tree.

My husband is retired, and spends his time outdoors hunting down these suckers and ripping them out. He even uses a dandelion weeder to pull out the roots! He has made such a mess of our backyard. It's full of lumps where he pulled out maple tree suckers. I will yell at him and tell him to stomp down the lumps, don't just leave them for me to trip over!!

Thanks for all of your advise. We will try cutting them off at the base of the tree once the snow is gone.