  We extend the classical Plateau-Rayleigh instability criterion in the
$\mathbb{E}(\kappa,\tau)$ spaces. We prove the existence of a positive number
$L_0>0$ such that if a truncated circular cylinder of radius $\rho$ in
$\mathbb{E}(\kappa,\tau)$ has length $L>L_0$ then it is unstable. This number
$L_0$ depends on $\kappa$, $\tau$ and $\rho$. The value $L_0$ is sharp under
axially-symmetric variations of the surface. We also extend this result for the
partitioning problem in $\mathbb{E}(\kappa,\tau)$.
