  It is well-known that the unit cotangent bundle of any Riemannian manifold
has a canonical contact structure. A surface in a Riemannian 3-manifold is
called a (wave) front if it is the projection of a Legendrian immersion into
the unit cotangent bundle. We shall give easily-computable criteria for a
singular point on a front to be a cuspidal edge or a swallowtail. Using this,
we shall prove that generically flat fronts in the hyperbolic 3-space admit
only cuspidal edges and swallowtails. Moreover, we will show that every
complete flat front (which is not rotationally symmetric) has associated
parallel surfaces whose singularities consist of only cuspidal edges and
swallowtails.
