  For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new
rigid and extremal effective codimension $j$ cycles in
$\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and
projections of these strata under forgetful morphisms and show the same holds
for $k$-differentials with $k\geq 3$ if the strata are irreducible. We compute
the class of the divisors in the case of quadratic differentials which contain
the first known examples of effective divisors on
$\overline{\mathcal{M}}_{g,n}$ with negative $\psi_i$ coefficients.
