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