  We investigate the classical stability of supersymmetric, asymptotically
flat, microstate geometries with five non-compact dimensions. Such geometries
admit an "evanescent ergosurface": a timelike hypersurface of infinite
redshift. On such a surface, there are null geodesics with zero energy relative
to infinity. These geodesics are stably trapped in the potential well near the
ergosurface. We present a heuristic argument indicating that this feature is
likely to lead to a nonlinear instability of these solutions. We argue that the
precursor of such an instability can be seen in the behaviour of linear
perturbations: nonlinear stability would require that all linear perturbations
decay sufficiently rapidly but the stable trapping implies that some linear
perturbation decay very slowly. We study this in detail for the most symmetric
microstate geometries. By constructing quasinormal modes of these geometries we
show that generic linear perturbations decay slower than any inverse power of
time.
