  We consider disordered Hamiltonians given by the Laplace operator subject to
arbitrary random self-adjoint singular perturbations supported on random
discrete subsets of the real line. Under minimal assumptions on the type of
disorder, we prove the following dichotomy: Either every realization of the
random operator has purely absolutely continuous spectrum or spectral and
exponential dynamical localization hold. In particular, we establish Anderson
localization for Schr\"odinger operators with Bernoulli-type random singular
potential and singular density.
