Hi David,
Thank you for your question about removing your aspen tree and eliminating the aspen shoots from your landscape. This is a very frustrating problem, and a common issue here in Central Oregon. Aspen roots are very vigorous. Aspens are genetically designed to be grove trees, so unfortunately, there is not much you can do to stop the roots from spreading.

Amy Jo Detweiler, a member of the OSU Extension horticulture staff here in Deschutes County, suggests the following for removing an aspen tree: “ If you are trying to remove an aspen and the suckers then you can cut down the main plant during the growing season, and then drill holes in the freshly cut stump and apply a brush remover herbicide (Crossbow) or products containing glyphosate (Roundup) this will help begin the process. If you have suckers coming up in a yard, you can paint Crossbow on the stems. Do this when temperatures are cooler (under 85 degrees) for herbicide use. Here is an article on it: az1401-08-14-06.indd (arizona.edu)”

If you consider carefully applying a non-selective herbicide such as Roundup on the shoots, it is important to note that the chemical does have the potential to kill other plants near the aspen roots or the lawn if the sprouts are in the grass.
The following link provides more detail about this process: Aspen Roots Showing | Forestry | Extension | USU.

If you have questions about the timing of your tree removal and potential damage to surrounding plants and the lawn, I recommend you call the Deschutes County OSU Extension office in Redmond at<personal data hidden>. Ask to speak with Amy Jo Detweiler (oregonstate.edu). Amy Jo welcomes your home garden questions and is a great resource. She may additional ideas for you.