  Photon circular orbits, an extreme case of light deflection, are among the
hallmarks of black holes and are known to play a central role in a variety of
phenomena related to these extreme objects. The very existence of such orbits
when motion is not confined in the equatorial plane, i.e. spherical orbits, is
indeed a special property of the separable Kerr metric and may not occur, for
instance, in the spacetime of other more speculative ultracompact objects. In
this paper we consider a general stationary-axisymmetric spacetime and examine
under what circumstances spherical or more general, variable-radius,
`spheroidal' non-equatorial photon orbits may exist with the ultimate goal of
using the modifications -- or even loss -- of photon trapping orbits as a
telltale of non-Kerr physics. In addressing this issue, we first derive a
general necessary condition for the existence of spherical/spheroidal orbits
and then go on to study photon trapping orbits in a variety of known non-Kerr
metrics (Johannsen, Johanssen-Psaltis, and Hartle-Thorne). The first of these
is an example of a separable spacetime which supports Kerr-like spherical
photon orbits. A more detailed analysis reveals a deeper connection between the
presence of spherical orbits and the separability of a metric (that is, the
existence of a third integral of motion). Specifically, a spacetime that does
not admit spherical orbits in any coordinates is necessarily non-separable. The
other two spacetimes considered here exhibit a clear non-Kerr behaviour by
having spherical photon orbits replaced by spheroidal ones. More importantly,
subject to the degree of deviation from Kerr, equatorial photon rings give
place to non-equatorial ones with an accompanying loss of low-inclination
spheroidal orbits. The implications of these results for the electromagnetic
and gravitational wave signature of non-Kerr objects are briefly discussed.
