  For each $n\in\mathbb{N}\cup\{\infty\}$, we introduce the notion of
$n$-singularity category $\mathbf{D}_{n{\rm-}sg}(R)$ of a given ring $R$, which
can be seen as a generalization of the classical singularity category.
Moreover, the $n$-global dimension $n$-gldim$(R)$ of $R$ is investigated. We
show that $\mathbf{D}_{n{\rm-}sg}(R)=0$ if and only if $n$-gldim$(R)$ is
finite. Furthermore, we characterize the vanishing property of $n$-singularity
categories in terms of recollements.
