  We study the Riemannian distance function from a fixed point (a point-wise
target) of Euclidean space in the presence of a compact obstacle bounded by a
smooth hypersurface. First, we show that such a function is locally semiconcave
with a fractional modulus of order one half and that, near the obstacle, this
regularity is optimal. Then, in the Euclidean setting, we prove that the
singularities of the distance function propagate, in the sense that each
singular point belongs to a nontrivial singular continuum. Finally, we
investigate the lack of differentiability of the distance function when a
convex obstacle is present.
