  Viable alternatives to astrophysical black holes include hyper-compact
objects without horizon, such as gravastars, boson stars, wormholes and
superspinars. The authors have recently shown that typical rapidly-spinning
gravastars and boson stars develop a strong instability. That analysis is
extended in this paper to a wide class of horizonless objects with approximate
Kerr-like geometry. A detailed investigation of wormholes and superspinars is
presented, using plausible models and mirror boundary conditions at the
surface. Like gravastars and boson stars, these objects are unstable with very
short instability timescales. This result strengthens previous conclusions that
observed hyper-compact astrophysical objects with large rotation are likely to
be black holes.
