Singularity Categories of B\"ackstr\"om Orders

B\"ackstr\"om orders are a class of algebras over complete discrete valuation rings. Their Cohen-Macaulay representations are in correspondence with the representations of certain quivers/species by Ringel and Roggenkamp. In this paper, we give explicit descriptions of the singularity categories of B\"ackstr\"om orders via certain von Neumann regular algebras and associated bimodules. We further provide singular equivalences between B\"ackstr\"om orders and specific finite dimensional radical square zero algebras. We also classify weakly regular, Gorenstein, Iwanaga-Gorenstein and sg-Hom-finite B\"ackstr\"om orders.