  For the distance function from any closed subset of any complete Finsler
manifold, we prove that the singular set is equal to a countable union of
delta-convex hypersurfaces up to an exceptional set of codimension two. In
addition, in dimension two, the whole singular set is equal to a countable
union of delta-convex Jordan arcs up to isolated points. These results are new
even in the standard Euclidean space and shown to be optimal in view of
regularity.
