  The time dependent Eikonal equation is a Hamilton-Jacobi equation with
Hamiltonian $H(P)=|P|$, which is not strictly convex nor smooth. The
regularizing effect of Hamiltonian for the Eikonal equation is much weaker than
that of strictly convex Hamiltonians, therefore leading to new phenomena such
as the appearance of "contact discontinuity". In this paper, we study the set
of singularity points of solutions in the upper half space for $C^1$ or $C^2$
initial data, with emphasis on the countability of connected components of the
set. The regularity of solutions in the complement of the set of singularity
points is also obtained.
