  We classify all cross-sections of Green's relations $\mathcal{L}$ and
$\mathcal{R}$ in the Brauer semigroup. The regular behavior of such
cross-sections starts from $n=7$. We show that in the regular case there are
essentially two different cross-sections and all others are
$\mathcal{S}_n$-conjugated to one of these two. We also classify all
cross-sections up to isomorphism.
