Optimal maps in $R^n$ to disconnected targets necessarily contain discontinuities (i.e.~tears). But how smooth are these tears? When the target components are suitably separated by hyperplanes, non-smooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions --- DC for short. If in addition the targets are convex the tears are actually $C^{1,\alpha}$. Similarly, under suitable affine independence assumptions, singularities of multiplicity $k$ lie on DC rectifiable submanifolds of dimension $n+1-k$. These are stable with respect to $W_\infty$ perturbations of the target measure. Moreover, there is at most one singularity of multiplicity $n$. This represents joint work with former Fields postdoc Jun Kitagawa (Michigan State University).