  We compute and explicitly describe the Bieri-Neumann-Strebel invariants
$\Sigma^1$ for the full and pure braid groups of the sphere $\mathbb{S}^2$, the
real projective plane $\mathbb{R}P^2$ and specially the torus $\mathbb{T}$ and
the Klein bottle $\mathbb{K}$. In order to do this for $M=\mathbb T$ or
$M=\mathbb K$, and $n \geq 2$, we use the $n^{th}$-configuration space of $M$
to show that the action by homeomorphisms of the group $Out(P_n(M))$ on the
character sphere $S(P_n(M))$ contains certain permutation of coordinates, under
which $\Sigma^1(P_n(\mathbb T))^c$ and $\Sigma^1(P_n(\mathbb K))^c$ are
invariant. Furthermore, $\Sigma^1(P_n(\mathbb T))^c$ and
$\Sigma^1(P_n(\mathbb{S}^2))^c$ (the latter with $n \geq 5$) are finite unions
of pairwise disjoint circles, and $\Sigma^1(P_n(\mathbb K))^c$ is finite. This
last fact implies that there is a normal finite index subgroup $H \leq
Aut(P_n(\mathbb K))$ such that the Reidemeister number $R(\varphi)$ is infinite
for every $\varphi \in H$.
