The presence of dry friction in mechanical systems induces the existence of an equilibrium set, consisting of infinitely many equilibrium points. The local dynamics of the trajectories near an equilibrium set is investigated for systems with one frictional interface. In this case, the equilibrium set will be an interval of a curve in phase space. It is shown in this paper that local bifurcations of equilibrium sets occur near the endpoints of this curve. Based on this result, sufficient conditions for structural stability of equilibrium sets in planar systems are given, and two new bifurcations are identified. The results are illustrated by application to a controlled mechanical system with friction.