I will discuss obstructions to metrics of positive scalar curvature with uniformly Euclidean point singularities. This provides counterexamples to a conjecture by Schoen. I will also discuss the existence of metrics with uniformly Euclidean point singularities which cannot be smoothed by a geometric flow while preserving nonnegativity of the scalar curvature. This is based on joint work with Georg Frenck and Rudi Zeidler