Hi Shari,  This is the million dollar question!  I spent yesterday in my own yard digging down deep into the soil to find the root and pull the root out where the sprouts are attached. This is harder to do in a lawn like setting. Having aspen suckers in the lawn is tough as there is not real easier way to eradicate them.  In theory if you were to use glyphosate (ingredient in Roundup) it would probably kill back the sprout but could go all the way back to the the mother plant, may not outright kill it but could weaken it. So that would be a risk you would have to take. Often in these cases the aspen is off site/property. This is the case with the suckers in my lawn they are from a sidewalk strip planting next to the house behind me.  Also any herbicide that would hit the lawn would kill the lawn as glyphosate is non-selective. I continue to try and follow the research on this dilema since aspen are heavily planted in Central Oregon. Currently now aware of any surefire way to kill the suckers without potentially harming the mother plant. The product Crossbow which contains 2,4-D and triclopyr is often recommended instead of glyphosate as it will not kill the grass. Please follow all label directions if you decide to use any herbicide. Not sure if Crossbow will harm the mother plant, I don't think the research is clear on this.  Thanks,