  We investigate stability properties of a type of periodic solutions of the
$N$-vortex problem on general domains $\Omega\subset \mathbb{R}^2$. The
solutions in question bifurcate from rigidly rotating configurations of the
whole-plane vortex system and a critical point $a_0\in\Omega$ of the Robin
function associated to the Dirichlet Laplacian of $\Omega$. Under a linear
stability condition on the initial rotating configuration, which can be
verified for examples consisting of up to 4 vortices, we show that the linear
stability of the induced solutions is solely determined by the type of the
critical point $a_0$. If $a_0$ is a saddle, they are unstable. Otherwise they
are stable in a certain linear sense. The proof uses a criterion for the
bifurcation of multiple eigenvalues, which is applied to suitable Poincar\'e
sections. Beyond linear stability, Herman's last geometric theorem allows us to
prove the existence of isoenergetically orbitally stable solutions in the case
of $N=2$ vortices.
