  We study the geometry of cuspidal $S_k$ singularities in $\mathbb R^3$
obtained by folding generically a cuspidal edge. In particular we study the
geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We
study geometrical invariants associated to $M$ and show that they determine it
up to order 5. We then study the flat geometry (contact with planes) of a
generic cuspidal cross-cap by classifying submersions which preserve it and
relate the singularities of the resulting height functions with the geometric
invariants.
