We prove Oort's conjecture that generically on the supersingular locus of the moduli space of principally polarized abelian varieties of genus g and in characteristic p, the automorphism group of the universal principally polarized abelian variety consists only of ±1, unless g=2 or 3 and p=2. On the way, we provide an explicit description of the a=1-locus in the Rapoport-Zink space of principally polarized supersingular p-divisible groups of any dimension g. We also prove analogous results for generic automorphism groups on moduli spaces of supersingular p-divisible groups with and without polarization.