  We consider in this paper discrete improper affine spheres based on
asymptotic nets. In this context, we distinguish the discrete edges and
vertices that must be considered singular. The singular edges can be considered
as discrete cuspidal edges, while some of the singular vertices can be
considered as discrete swallowtails. The classification of singularities of
discrete nets is a quite difficult task, and our results can be considered as a
fisrt step in this direction. We also prove some characterizations of ruled
discrete improper affine spheres which are analogous to the smooth case.
