How to Identify and Remove Rose Suckers
Suckers are shoots growing from the rootstock below a grafted rose's bud union, not from the variety itself. Remove them by tearing (not cutting) them off at the root where they emerge, as soon as you spot them.
Many roses are grafted: the desirable variety is budded onto a vigorous rootstock. Sometimes the rootstock sends up its own shoots — suckers — from below the bud union. Left alone, they can overtake and replace the variety you planted. Tell-tale signs are foliage that looks different from the rest of the plant (often lighter, with a different leaf-count) and shoots emerging from below the graft or from the surrounding soil.
To remove a sucker, trace it down to where it joins the root and tear it off rather than cutting — cutting leaves dormant buds that regrow. Removing suckers promptly keeps the plant's energy in the variety you want. Roses grown on their own roots never sucker in this way.