We investigate numerically the quasiparticle density of states ϱ(E) for a two-dimensional, disordered superconductor in which both time-reversal and spin-rotation symmetries are broken. As a generic single-particle description of this class of systems (symmetry class D), we use the Cho-Fisher version of the network model. This has three phases: a thermal insulator, a thermal metal, and a quantized thermal Hall conductor. In the thermal metal, we find a logarithmic divergence in ϱ(E) as E→0, as predicted from sigma model calculations. Finite-size effects lead to superimposed oscillations, as expected from random-matrix theory. In the thermal insulator and quantized thermal Hall conductor, we find that ϱ(E) is finite at E=0. At the plateau transition between these phases, ϱ(E) decreases toward zero as ∣E∣ is reduced, in line with the result ϱ(E)∼∣E∣ln(1∕∣E∣) derived from calculations for Dirac fermions with random mass.