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We study a moduli space ${\mathrm{𝒜𝒮}}_g$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$. We study the stratification of ${\mathrm{𝒜𝒮}}_g$ by $p$-rank into strata ${\mathrm{𝒜𝒮}}_{g.s}$ of Artin-Schreier curves of genus $g$ with $p$-rank exactly $s$. We enumerate the irreducible components of ${\mathrm{𝒜𝒮}}_{g,s}$ and find their dimensions. As an application, when $p=2$, we prove that every irreducible component of the moduli space of hyperelliptic $k$-curves with genus $g$ and $2$-rank $s$ has dimension $g-1+s$. We also determine all pairs $(p,g)$ for...