  We prove that for all integers $r \geq 2$ and $g \geq \lfloor
\frac{r^2+10r+1}{4} \rfloor$ there exists a component of the locus
$\mathcal{S}^r_g$ of spin curves with a theta characteristic $L$ such that
$h^0(L) \geq r+1$ and $h^0(L)\equiv r+1 (\text{mod} 2)$ which has expected
codimension $\binom{r+1}{2}$ inside the moduli space $\mathcal{S}_g$ of spin
curves of genus $g$.
