Lilacs are tough shrubs, tolerant of cold and drought, experts at survival. Part of their strategy is sending out suckers, shoots from the roots or the base of the bush, creating a thicket of stems that can be as much as 10 feet in diameter. If you want to confine it to a smaller area or kill the bush completely, you'll have to kill the roots themselves since their ability to regenerate is considerable.
Most people wish to limit the size of their bush, not kill it outright, so chemicals that move through the stems into the roots--translocated brush killers--cannot be used without severing the connection between the shoots and the main part of the plant.
Simple physical removal of roots by digging, though effective, can be too costly in terms of time and energy and will disturb other plants in the bed. Some varieties of lilac, such as 'Palibin', sucker less than others and planting these may avoid the problem or at least reduce the shrub's tendency to spread.
A chemical product used in the orchard industry to prevent suckering from roots and pruning cuts is available for homeowners, according to home improvement expert Danny Lipford. Called "Monterey Sucker Stopper," it contains the growth inhibitor NAA (napthaleneacetate).
Other root killing options include using a double-thick layer of plastic covered with mulch at the far edges of a lilac bush so the main stems continue to receive water; digging a narrow trench around the bush at the the edge of the leaf canopy and using a translocated brushkiller to kill shoots beyond the trench; or installing metal or plastic edging, 6 to 12 inches deep, as a physical barrier to keep the roots from expanding beyond your desired size.
The effects of a sucker-preventing chemical are limited to the surface of the root or pruning cut. Translocated brush killers destroy all growth, both roots and stems. Cutting roots and pulling shoots is a temporary fix that can actually encourage sucker production as a stress response. A physical barrier that restricts root growth causes a greater density of roots within that block.
Limiting the root growth of a lilac allows you keep it at size appropriate for your space. Using a sucker-preventing chemical allows the roots to continue to expand, nourishing the plant. A physical barrier is perhaps the most permanent solution, saving time and effort in the long run.
A sucker-preventing chemical may need to be applied repeatedly as the root ball expands. Plastic sheeting breaks down over time and may need to be replaced. Its use is incompatible with growing plants nearby. Root pruning will need to be repeated as roots grow. Translocated brushkillers can be toxic to surrounding plants. Restricting roots inside a physical barrier can dwarf the lilac and cause a need for more frequent fertilization.