  We consider the problem of describing the asymptotic behaviour of
\textsc{FRW} universes near their spacetime singularities in general
relativity. We find that the Bel-Robinson energy of these universes in
conjunction with the Hubble expansion rate and the scale factor proves to be an
appropriate measure leading to a complete classification of the possible
singularities. We show how our scheme covers all known cases of cosmological
asymptotics possible in these universes and also predicts new and distinct
types of singularities. We further prove that various asymptotic forms met in
flat cosmologies continue to hold true in their curved counterparts. These
include phantom universes with their recently discovered big rips, sudden
singularities as well as others belonging to graduated inflationary models.
