As a stepping stone toward a theory of positive scalar curvature for C^0 metrics, Gromov has proposed a framework of polyhedral comparisons that hinges on understanding which smooth obstructions to positive scalar curvature carry through to settings where specific singularities are allowed. In this talk we discuss known and conjectured obstructions to positive scalar curvature in the presence of such singular sets with codimensions 1, 2, and 3 or higher. This is joint work with Chao Li.