  Let $\bar{Y}$ be a normal surface that is the canonical $\mu_2$- or
$\alpha_2$-covering of a classical or supersingular Enriques surface in
characteristic $2$. We determine all possible configurations of singularities
on $\bar{Y}$, and for each configuration we describe which type of Enriques
surfaces (classical or supersingular) appear as quotients of $\bar{Y}$.
