We give general results for the asymptotics and anomaly for volumes of regions with respect to measures that are singular along hypersurfaces. These have applications to holography in physical models as well as invariant theory for embedded hypersurfaces. In both cases one solves an appropriate bulk problem with boundary data specified along the singular hypersurface. In particular we focus on applications to the conformal geometry of hypersurfaces where the appropriate bulk probem is a singular version of the classical Yamabe problem of finding conformal metric rescalings that bring the scalar curvature to a constant value.