Continuing with my own version of the Failed States Index (see here for background and methods), here we have the Europe map (including Caucasus, Cyprus and Turkey but not Central Asia):
Notice that I may have used somewhat different criteria in the cases of Serbia, Georgia, Azerbaijan, Moldova and Cyprus, all them affected by the formation of separate ethnic states by means of foreign intervention. In the cases of Azerbaijan and Georgia I considered the conflicts opened and the states subject to foreign military intervention (by Armenia and Russia respectively), in the cases of Moldova and Cyprus I considered the conflict contained and the states subject to foreign military intervention (by Russia and Turkey respectively), in the case of Serbia I considered Kosovo as a de facto separate state (even if Serbia is reluctant to accept the status quo, it is not likely that will do or even will be able to do anything about it either). These decisions, admittedly subjective and arguable, have probably affected to some extent the stability index of these countries (re-evaluate at will).
We can in any case say that Serbia without Kosovo is more stable than with it. Even if it still has latent ethnic conflict in Vojvodina and a small town at the Kosovan border, this is not enough to damage its stability overall. But with Kosovo it'd be quite different: Serbia would be more unstable, possibly even as much as Georgia.
And vice versa: Georgia might want to sign peace with its breakaway nations and Russia in order to improve its stability, though this would also ask for greater democracy and income distribution.
Beside these peculiar cases, what I see is that two factors are crucial in Europe: ethnic homogeneity and to a lesser extent foreign military presence (typically as NATO bases). I understand that military bases equate to a mild foreign occupation and show that a state is either too weak to defend itself on its own means or too weak to put down the demands of imperialist powers.
But the crucial element seems to be open ethnic conflict, which we find in Spain, Turkey, Russia and the other states mentioned above.
Still there are not obvious failed states in Europe but there are some in the Caucasus that are close and others that are weak enough to be blown to pieces by an intensification of the instability currents eventually.