  A wide range of equations related to free surface motion in two dimensions
exhibit the formation of cusp singularities either in time, or as function of a
parameter. We review a number of specific examples, relating in particular to
fluid flow and to wave motion, and show that they exhibit one of two types of
singularity: cusp or swallowtail. This results in a universal scaling form of
the singularity, and permits a tentative classification.
