  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 show that the
distance function is everywhere differentiable (except for the point-wise
target) if and only if no obstacle is present. Finally, we prove that all the
singular points of the distance function are not isolated, in the sense that
each singularity belongs to a nontrivial continuum of singular points.
