  We present a method for computing $\mathbb{A}^1$-homotopy invariants of
singularity categories of rings admitting suitable gradings. Using this we
describe any such invariant, e.g. homotopy K-theory, for the stable categories
of self-injective algebras admitting a connected grading. A remark is also made
concerning the vanishing of all such invariants for cluster categories of type
$A_{2n}$ quivers.
