Over a field of positive characteristic p, we consider moduli spaces of polarized abelian varieties equipped with an action by a ring unramified at p. Using deformation theory, we show that ordinary points are dense in each of the following situations: the polarization is separable; the polarization is mildly inseparable, and the ring of endomorphisms is a totally real number field; or the polarization is arbitrary, and the ring is a real quadratic field acting on abelian fourfolds. We introduce a new invariant which measures the extent to which a polarized Dieudonne module admits an isotropic splitting lifting the Hodge filtration, and use it to explain the singularities arising from mildly inseparable polarizations
