  We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb
S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with
$n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at
least $\frac{n}{2}+1$ that satisfies the wrapping property. The notion of
wrapping property was introduced by the second author for studying related
scalar curvature rigidity problems on spheres. For example, any closed subset
of $\mathbb S^n$ contained in a hemisphere and any finite subset of $\mathbb
S^n$ satisfy the wrapping property. The same techniques also apply to prove an
analogous scalar rigidity result for $L^\infty$ metrics on tori that are smooth
away from certain subsets of codimension at least $\frac{n}{2}+1$.
