  Main purpose of this paper is to study the local propagation of singularities
of viscosity solution to contact type evolutionary Hamilton-Jacobi equation $$
D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0. $$ An important issue of this topic is the
existence, uniqueness and regularity of the strict singular characteristic. We
apply the recent existence and regularity results on the Herglotz' type
variational problem to the aforementioned Hamilton-Jacobi equation with smooth
initial data. We obtain some new results on the local structure of the cut set
of the viscosity solution near non-conjugate singular points. Especially, we
obtain an existence result of smooth strict singular characteristic from and to
non-conjugate singular initial point based on the structure of the
superdifferential of the solution, which is even new in the classical
time-dependent case. We also get a global propagation result for the $C^1$
singular support in the contact case.
