Calculations of the densities of states of the Anderson-model Hamiltonian are reported for cases with diagonal and off-diagonal disorder and for the following lattices: in two dimensions, (i) simple square, (ii) honeycomb, and (iii) triangular; and in three dimensions, simple cubic. In all cases diagonal disorder destroys the van Hove singularity peaks. With strong off-diagonal disorder, the density of states for a square lattice exhibits a band-center peak, at E=0, where the ordered lattice had a van Hove singularity; no such peak is found for a triangular lattice. These results correlate with the conclusion of Soukoulis et al. for off-diagonal disorder that the E=0 state of the triangular lattice is exponentially localized, but that of the square lattice is not. However, zero-energy peaks are found for both the honeycomb two-dimensional lattice and the cubic three-dimensional lattice, demonstrating that the band-center peak is not derived from a van Hove singularity of the ordered Hamiltonian. The implications are discussed.