  We give a new and detailed description of the structure of cut loci, with
direct applications to the singular sets of some Hamilton-Jacobi equations.
These sets may be non-triangulable, but a local description at all points
except for a set of Hausdorff dimension $n-2$ is well known. We go further in
this direction by giving a clasification of all points up to a set of Hausdorff
dimension $n-3$.
