From the ArcaMax Publishing, Gardening Newsletter:
http://www.arcamax.com/news/gardening/s-1887-311373
Question: I have several lilacs in my yard, and each year they send up
lots of shoots around their bases, and are getting kind of messy
looking. Should I be pruning these off? Is there anything I can do to
prevent them from forming?
Answer: Lilacs naturally sucker, or send up little runners or shoots.
They do this as a method of self renewal. The recommended annual
pruning includes removing any suckers smaller than a pencil by cutting
them at the base. It also includes removing a portion of the oldest
stems by cutting them at the base. This way the plant has a steady
influx of vigorous new wood and will not become tall and gangly.