  We prove that the moduli space of double covers ramified at two points
$\mathcal{R}_{g,2}$ is uniruled for $3\leq g\leq 6$ and of general type for
$g\geq 16$. Furthermore, we consider Prym-canonical divisorial strata in the
moduli space $\overline{\mathcal{C}^n\mathcal{R}}_g$ parametrizing $n$-pointed
Prym curves, and we compute their classes in
$\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{C}^n\mathcal{R}}_g)$.
