When are splitting loci Gorenstein?

Splitting loci are certain natural closed substacks of the stack of vector bundles on $\mathbb{P}^1$, which have found interesting applications in the Brill-Noether theory of $k$-gonal curves. In this paper, we completely characterize when splitting loci, as algebraic stacks, are Gorenstein or $\mathbb{Q}$-Gorenstein. The main ingredients of the proof are a computation of the class groups of splitting loci in certain affine extension spaces, and a formula for the class of their canonical module.