In this talk I will first review the recent progress made towards finding instabilities of rotating vacuum black holes in higher dimensions. I will show that singly spinning Myers-Perry black holes exhibit an infinite sequence of zero modes that should connect them to other black hole phases such as black rings and black saturns. In the second part of my talk I will describe the gravitational instabilities of Myers-Perry black holes with all the angular momenta equal. The onset of instability is associated with the appearance of time-independent perturbations which generically break all but one of the rotational symmetries. In nine spacetime dimensions, this is interpreted as evidence for the existence of a new 70-parameter family of black hole solutions with only a single rotational symmetry.