Not yet reviewed by Pith; the record is open.

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.