  Let $K$ be a finite unramified extension of $\mathbb{Q}_p$ with $p > 5$. We
study the extremely non-generic irreducible components in the reduced part of
the Emerton-Gee stack for $\mathrm{GL}_2$. We show precisely which irreducible
components are smooth or normal, and which have Gorenstein or Cohen-Macaulay
normalizations, as well as give a description of the singular loci. We use our
results to update expectations about the conjectural categorical $p$-adic
Langlands correspondence.
