Propagation of singularities for solutions to Hamilton-Jacobi equations

The study of the structural properties of the set of points at which the viscosity solution of a first order Hamilton-Jacobi equation fails to be differentiable - in short, the singular set - started with the paper [On the singularities of viscosity solutions to Hamilton-Jacobi-Bellman equations, Indiana Univ. Math. J. 36 (1987), pp.501-524] by Mete Soner and myself. These thirty years have registered enormous progress in the comprehension of the way how singularities propagate: a fine measure theoretical analysis of the singular set has been developed, it has been understood how singular dynamics is driven by generalised characteristics, and deep topological applications to the structure of the cut locus on a Riemannian manifold have been pointed out. In this talk, I will revisit the milestones of the theory and discuss possible developments and open problems.