  We first analyze the integrated density of states (IDS) of periodic
  Schr\"odinger operators on an amenable covering manifold. A criterion for the
continuity of the IDS at a prescribed energy is given along with examples of
operators with both continuous and discontinuous IDS'. Subsequently, alloy-type
perturbations of the periodic operator are considered. The randomness may enter
both via the potential and the metric. A Wegner estimate is proven which
implies the continuity of the corresponding IDS. This gives an example of a
discontinuous "periodic" IDS which is regularized by a random perturbation.
