  In this paper, we give the maximum of the numbers $n$ such that we can take
$n$ simple closed geodesics without singularities that are disjoint to each
other for translation surfaces in the hyperelliptic components
$\mathcal{H}^{\rm {hyp}}(2g-2)$ and $\mathcal{H}^{\rm{hyp}}(g-1, g-1)$. The
maximum is different from that of the case of hyperbolic surfaces. We also give
a classification theorem for translation surfaces in
$\mathcal{H}^{\mathrm{hyp}}(4)$ with respect to their Euclidean structures.
