  Wide-field optical imaging surveys will contain tens of thousands of new
strong gravitational lenses. Some of these will have new and unusual image
configurations, and so will enable new applications: for example, systems with
high image multiplicity will allow more detailed study of galaxy and group mass
distributions, while high magnification is needed to super-resolve the faintest
objects in the high redshift universe. Inspired by a set of six unusual lens
systems [including five selected from the Sloan Lens ACS (SLACS) and Strong
Lensing Legacy (SL2S) surveys, plus the cluster Abell 1703], we consider
several types of multi-component, physically-motivated lens potentials, and use
the ray-tracing code "glamroc" to predict exotic image configurations. We also
investigate the effects of galaxy source profile and size, and use realistic
sources to predict observable magnifications and estimate very approximate
relative cross-sections. We find that lens galaxies with misaligned disks and
bulges produce swallowtail and butterfly catastrophes, observable as "broken"
Einstein rings. Binary or merging galaxies show elliptic umbilic catastrophes,
leading to an unusual Y-shaped configuration of 4 merging images. While not the
maximum magnification configuration possible, it offers the possibility of
mapping the local small-scale mass distribution. We estimate the approximate
abundance of each of these exotic galaxy-scale lenses to be ~1 per all-sky
survey. In higher mass systems, a wide range of caustic structures are
expected, as already seen in many cluster lens systems. We interpret the
central ring and its counter-image in Abell 1703 as a "hyperbolic umbilic"
configuration, with total magnification ~100 (depending on source size). The
abundance of such configurations is also estimated to be ~1 per all-sky survey.
