  One way to understand more about spacetime singularities is to construct
solutions of the Einstein equations containing singularities with prescribed
properties. The heuristic ideas of the BKL picture suggest that oscillatory
singularities should be very common and give a detailed picture of how these
could look. The more straightforward case of singularities without oscillations
is reviewed and existing results on that subject are surveyed. Then recent
theorems proving the existence of spatially homogeneous solutions with
oscillatory singularities of a specific type are presented. The proofs of these
involve applications of some ideas concerning heteroclinic chains and their
stability. Some necessary background from the theory of dynamical systems is
explained. Finally some directions in which this research might be generalized
in the future are pointed out.
