  We prove, for infinitely many values of $g$ and $n$, the existence of
non-tautological algebraic cohomology classes on the moduli space
$\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular,
when $n=0$, our results show that there exist non-tautological algebraic
cohomology classes on $\mathcal{M}_g$ for $g=12$ and all $g \geq 16$. These
results generalize the work of Graber--Pandharipande and van Zelm, who proved
that the classes of particular loci of bielliptic curves are non-tautological
and thereby exhibited the only previously-known non-tautological class on any
$\mathcal{M}_g$: the bielliptic cycle on $\mathcal{M}_{12}$. We extend their
work by using the existence of holomorphic forms on certain moduli spaces
$\overline{\mathcal{M}}_{g,n}$ to produce non-tautological classes with
nontrivial restriction to the interior, via which we conclude that the classes
of many new double-cover loci are non-tautological.
