  Caustics are a striking phenomena in natural optics and hydrodynamics:
high-amplitude characteristic patterns that are singular in the short
wavelength limit. We use exact numerical and approximate semiclassical analytic
methods to study quantum versions of caustics that form in Fock space during
far-from-equilibrium dynamics following a quench in the Bose Hubbard dimer and
trimer models. Due to interparticle interactions, both models are nonlinear and
the trimer is also nonintegrable. The caustics exhibit a hierarchy of
morphologies described by catastrophe theory that are stable against
perturbations and hence occur generically. The dimer case gives rise to
discretized versions of folds and cusps which are the simplest two of Thom's
catastrophes. The extra dimension available in the trimer case allows for
higher catastrophes including the codimension-3 hyperbolic umbilic and elliptic
umbilic catastrophes which we argue are organized by, and projections of, an
8-dimensional corank-2 catastrophe known as $X_9$. These results indicate a
hitherto unrecognized form of universality in quantum many-body dynamics
organized by singularities.
