  An $N$-channel spinless p-wave superconducting wire is known to go through a
series of $N$ topological phase transitions upon increasing the disorder
strength. Here, we show that at each of those transitions the density of states
shows a Dyson singularity $\nu(\varepsilon) \propto
\varepsilon^{-1}|\ln\varepsilon|^{-3} $, whereas $\nu(\varepsilon) \propto
\varepsilon^{|\alpha|-1}$ has a power-law singularity for small energies
$\varepsilon$ away from the critical points. Using the concept of
"superuniversality" [Gruzberg, Read, and Vishveshwara, Phys. Rev. B 71, 245124
(2005)], we are able to relate the exponent $\alpha$ to the wire's transport
properties at zero energy and, hence, to the mean free path $l$ and the
superconducting coherence length $\xi$.
