For a class of hybrid systems given in terms of constrained differential and difference equations/inclusions, we study the existence of control Lyapunov functions when compact sets are asymptotically stable as well as the stabilizability properties guaranteed when control Lyapunov functions exist. An existence result asserting that asymptotic stabilizability of a compact set implies the existence of a smooth control Lyapunov function is established. When control Lyapunov functions are available, conditions guaranteeing the existence of stabilizing continuous state-feedback control laws are provided.