Some types of trees are more prone than others to produce this sort of basal growth. Lime trees (Tilia species) are particularly well known for it. We used to have the same growth from sycamores. All I know to do with it is just cut off the shoots as they appear, or once a year if you don't keep an eye on them all the time.

Beattie's right, there's no way of stopping them - its at least an annual job, if not more often.

Easiest to keep an eye on them, and put on the garden gloves and rub them off before they get more than a cm long. That often removes the buds at the base of the shoots, which reduces the next crop of suckers. If you are talking about shoots coming from below ground, that's a different story!