  For a higher Nakayama algebra $A$ in the sense of Jasso-K\"{u}lshammer, we
show that the singularity category of $A$ is triangulated equivalent to the
stable module category of a self-injective higher Nakayama algebra. This
generalizes a similar result for usual Nakayama algebras due to Shen. Our proof
relies on the existence of $d\mathbb{Z}$-cluster tilting subcategories in the
module category of $A$ and the result of Kvamme that each $d\mathbb{Z}$-cluster
tilting subcategory of $A$ induces a $d\mathbb{Z}$-cluster tilting subcategory
in its singularity category. Moreover, our result provides many concrete
examples of the triangulated Auslander-Iyama correspondence introduced by
Jasso-Muro, namely, there is a bijective correspondence between the equivalence
classes of the singularity categories of $d$-Nakayama algebras with its basic
$d\mathbb{Z}$-cluster tilting object and the isomorphism classes of
self-injective $(d+1)$-Nakayama algebras.
