  For large random matrices $X$ with independent, centered entries but not
necessarily identical variances, the eigenvalue density of $XX^*$ is
well-approximated by a deterministic measure on $\mathbb{R}$. We show that the
density of this measure has only square and cubic-root singularities away from
zero. We also extend the bulk local law in [arXiv:1606.07353] to the vicinity
of these singularities.
