  If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of
the evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$
where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli
Hamiltonian, we prove the set $\Sigma(U)$, of points where $U$ is not
differentiable, is locally contractible. Moreover, we study the homotopy type
of $\Sigma(U)$. We also give an application to the singularities of a distance
function to a closed subset of a complete Riemannian manifold.
