  $n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher
dimensional Auslander-Reiten theory. We give a complete classification of
$n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama
algebras. In particular, we show that there are three kinds of Nakayama
algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global
dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective
ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has
been shown by the second author, that each such $n\mathbb{Z}$-cluster tilting
subcategory induces an $n\mathbb{Z}$-cluster tilting subcategory of the
corresponding singularity category. For each Nakayama algebra in our
classification, we describe its singularity category, the canonical functor
from its module category to its singularity category, and provide a complete
comparison of $n\mathbb{Z}$-cluster tilting subcategories in the module
category and the singularity category. This relies heavily of results by Shen,
who described the singularity categories of all Nakayama algebras.
