The normal modes of a sequence of two-dimensional flows around a black hole are investigated. When the flow is one of pure rotation, numerous non-axisymmetric dynamical instabilities are found which are similar to those discovered by previous authors. When the inner edge of the flow crosses the critical ‘cusp’ radius, a purely rotating equilibrium flow is no longer possible and the fluid in the inner regions flows into the hole transonically. All the unstable modes found for the purely rotating flow are quickly stabilized by this process. The relevance of these results to the dynamical stability of accreting thick tori is discussed.