  This is a review of recent progress concerning generic spacelike
singularities in general relativity. For brevity the main focus is on
singularities in vacuum spacetimes, although the connection with, and the role
of, matter for generic singularity formation is also commented on. The paper
describes recent developments in two areas and show how these are connected
within the context of the conformally Hubble-normalized state space approach.
The first area is oscillatory singularities in spatially homogeneous cosmology
and the connection between asymptotic behaviour and heteroclinic chains. The
second area concerns oscillatory singularities in inhomogeneous models,
especially spike chains and recurring spikes. The review also outlines some
underlying reasons for why the structures that are the foundation for generic
oscillatory behaviour exists at all, which entails discussing how underlying
physical principles and applications of solution generating techniques yield
hierarchical structures and connections between them. Finally, it is pointed
out that recent progress concerning generic singularities motivates some
speculations that suggest that a paradigm shift concerning their physical role,
and what mathematical issues to address, might be in order.
