  There are only three stable singularities of a differentiable map between
three-dimensional manifolds, namely folds, cusps and swallowtails. A Skyrme
configuration is a map from space to SU(2), and its singularities correspond to
the points where the baryon density vanishes. In this paper we consider the
singularity structure of Skyrme configurations. The Skyrme model can only be
solved numerically. However, there are good analytic ansaetze. The simplest of
these, the rational map ansatz, has a non-generic singularity structure. This
leads us to introduce a non-holomorphic ansatz as a generalization. For baryon
number two, three and four, the approximate solutions derived from this ansatz
are closer in energy to the true solutions than any other ansatz solution. We
find that there is a tiny amount of negative baryon density for baryon number
three, but none for two or four. We comment briefly on the relationship to
Bogomolny-Prasad-Sommerfield monopoles.
