We consider the stability of black holes within both classical general relativity and the semiclassical thermodynamic description. In particular, we study linearised perturbations and their contribution to the gravitational partition function, addressing technical issues for charged (Reissner-Nordstrom) and rotating (Kerr-AdS) black holes. Exploring the connection between classical and thermodynamic stability, we find classical instabilities of Myers-Perry black holes and bifurcations to new black hole families.