Non-generic components of the Emerton-Gee stack for GL₂

Let K be a finite unramified extension of Qₚ with p > 5. We study the extremely non-generic irreducible components in the reduced part of the Emerton-Gee stack for GL₂. 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.