Viscosity solutions of Hamilton-Jacobi-Bellman equations are nonsmooth functions which may fail to be differentiable on small sets. Such singularities, which play an important role for the underlying optimal control problem, have been analyzed from various viewpoints. Their dynamics can be described by generalized characteristics, which are forward solutions of the charactheristic system in Filippov's sense. In this talk, for stationary Tonelli Hamiltonians we develop an intrinsic proof of the existence of generalized characteristics using the positive Lax-Oleinik semigroup. This approach brings to light the topological structure of the singular set of a viscosity solution, which turns out to be locally path connected as shown in a recent joint work with W. Cheng and A. Fathi.