  Caustics occur widely in dynamics and take on shapes classified by
catastrophe theory. At finite wavelengths they produce interference patterns
containing networks of vortices (phase singularities). Here we investigate
caustics in quantized fields, focusing on the collective dynamics of quantum
spins. We show that, following a quench, caustics are generated in the Fock
space amplitudes specifying the many-body configuration and which are
accessible in experiments with cold atoms, ions or photons. The granularity of
quantum fields removes all singularities, including phase singularities,
converting point vortices into nonlocal vortices that annihilate in pairs as
the quantization scale is increased. Furthermore, the continuous scaling laws
of wave catastrophes are replaced by discrete versions. Such `quantum
catastrophes' are expected to be universal dynamical features of quantized
fields.
