  This paper studies the structure of the singular set (points of
nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations
associated with general mechanical systems on the n-torus. First, using the
level set method, we characterize the propagation of singularities along
generalized characteristics. Then, we obtain a local propagation result for
singularities of weak KAM solutions in the supercritical case. Finally, we
apply such a result to study the propagation of singularities for barrier
functions.
