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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 themoduli space $\overline{\mathcal{C}^n\mathcal{R}}_g$ parametrizing $n$-pointedPrym curves, and we compute their classes in$\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{C}^n\mathcal{R}}_g)$.

The birational geometry of $\overline{\mathcal{R}}_{g,2}$ and Prym-canonical divisorial strata