What to do when lilac bush is growing suckers
We can see the main branches of our lilac bush (this is only our 2nd spring at this house), but under the main bush/tree tons of lilac leave suckers are growing up and into the grass & surrounding mulch beds.
There are a few options. The suckers can be managed by using a product called Sucker Stopper (or simmiliary named product) that will not hurt the main plant. There appears to be dead canes in the center of the bush. Removal of the dead plant material and then performing a rejuvenating prune (cutting the plant down to about 18" off the ground) may help revitalize the shrub to a more pleasing form. After a hard prune it will take the plant several years before it flowers again.
Or they can focus on removing of the sucker undergrowth this year and prune out the dead canes in the center and see how it responds.
Unfortunately sucker development is common with some trees and shrubs. There is no simple, easy way to control them. Since they are connected to the main plant, if you spray them with a systemic product like RoundUp (glyphosate) it will move into and potentially kill the whole shrub. You can spray them with non-systemic, contact herbicides like Finale (glufosinate) or diquat, but the suckers will continue to grow back. Products like Bonide Sucker Punch and Sucker Stopper (ethyl-naphthaleneacetate) also provides very limited, if any, control.
You could dig up any suckers growing in areas where you don't want them. But you'll need to keep doing that each year, since they will want to continue to spread. There is no permanent, long-term control.