  It has been recently shown, that some Skyrme functionals can lead to
non-converging results in the calculation of some properties of atomic nuclei.
A previous study has pointed out a possible link between these convergence
problems and the appearance of finite-size instabilities in symmetric nuclear
matter (SNM) around saturation density.
  We show that the finite-size instabilities not only affect the ground state
properties of atomic nuclei, but they can also influence the calculations of
vibrational excited states in finite nuclei. We perform systematic fully-self
consistent Random Phase Approximation (RPA) calculations in spherical
doubly-magic nuclei. We employ several Skyrme functionals and vary the
isoscalar and isovector coupling constants of the time-odd term
$\mathbf{s}\cdot \Delta \mathbf{s}$ . We determine critical values of these
coupling constants beyond which the RPA calculations do not converge because
RPA the stability matrix becomes non-positive.By comparing the RPA calculations
of atomic nuclei with those performed for SNM we establish a correspondence
between the critical densities in the infinite system and the critical coupling
constants for which the RPA calculations do not converge. We find a
quantitative stability criterion to detect finite-size instabilities related to
the spin $\mathbf{s}\cdot \Delta \mathbf{s}$ term of a functional. This
criterion could be easily implemented into the standard fitting protocols to
fix the coupling constants of the Skyrme functional.
