  We show that in every dimension $n \geq 8$, there exists a smooth closed
manifold $M^n$ which does not admit a smooth positive scalar curvature ("psc")
metric, but $M$ admits an $\mathrm{L}^\infty$-metric which is smooth and has
psc outside a singular set of codimension $\geq 8$. This provides
counterexamples to a conjecture of Schoen. In fact, there are such examples of
arbitrarily high dimension with only single point singularities. In addition,
we provide examples of $\mathrm{L}^\infty$-metrics on $\mathbb{R}^n$ for
certain $n \geq 8$ which are smooth and have psc outside the origin, but cannot
be smoothly approximated away from the origin by everywhere smooth Riemannian
metrics of non-negative scalar curvature. This stands in precise contrast to
established smoothing results via Ricci--DeTurck flow for singular metrics with
stronger regularity assumptions. Finally, as a positive result, we describe a
$\mathrm{KO}$-theoretic condition which obstructs the existence of
$\mathrm{L}^\infty$-metrics that are smooth and of psc outside a finite subset.
This shows that closed enlargeable spin manifolds do not carry such metrics.
