  We describe a new class of nonequilibrium quantum many-body phenomena in the
form of networks of caustics that dominate the many-body wavefunction in the
semiclassical regime following a sudden quench. It includes the light cone-like
propagation of correlations as a particular case. Caustics are singularities
formed by the birth and death of waves and form a hierarchy of universal
patterns whose natural mathematical description is via catastrophe theory.
Examples in classical waves range from rainbows and gravitational lensing in
optics to tidal bores and rogue waves in hydrodynamics. Quantum many-body
caustics are discretized by second-quantization (``quantum catastrophes'') and
live in Fock space which can potentially have many dimensions. We illustrate
these ideas using the Bose Hubbard dimer and trimer models which are simple
enough that the caustic structure can be elucidated from first principles and
yet run the full range from integrable to nonintegrable dynamics. The dimer
gives rise to discretized versions of fold and cusp catastrophes whereas the
trimer allows for higher catastrophes including the codimension-3 hyperbolic
and elliptic umbilics which are organized by, and projections of, an
8-dimensional corank-2 catastrophe known as $X_9$. These results describe a
hitherto unrecognized form of universality in quantum dynamics organized by
singularities that manifest as strong fluctuations in mode population
probabilities.
