We compute the exact log-asymptotics of the persistence probability, and determine the entropic repulsion profile conditioned on persistence, for general $d$-dimensional stationary Gaussian fields with spectral singularity at the origin of order $α\in [0,d)$. Under mild regularity conditions these are shown to be universal, depending only on $α$ and $d$, and to have explicit formulations in terms of the capacity and equilibrium potential of the $α$-Riesz kernel. This generalises a result of Bolthausen, Deuschel and Zeitouni on the Gaussian free field to a wide class of Gaussian fields with spectral singularity.