  We analyze (3+1)-dimensional black-hole space-times in spontaneously broken
Yang-Mills gauge theories that have been recently presented as candidates for
an evasion of the scalar-no-hair theorem. Although we show that in principle
the conditions for the no-hair theorem do not apply to this case, however we
prove that the `spirit' of the theorem is not violated, in the sense that there
exist instabilities, in both the sphaleron and gravitational sectors. The
instability analysis of the sphaleron sector, which was expected to be unstable
for topological reasons, is performed by means of a variational method. As
shown, there exist modes in this sector that are unstable against linear
perturbations. Instabilities exist also in the gravitational sector. A method
for counting the gravitational unstable modes, which utilizes a
catastrophe-theoretic approach is presented. The r\^ole of the catastrophe
functional is played by the mass functional of the black hole. The Higgs vacuum
expectation value (v.e.v.) is used as a control parameter, having a critical
value beyond which instabilities are turned on. The (stable) Schwarzschild
solution is then understood from this point of view. The catastrophe-theory
appproach facilitates enormously a universal stability study of non-Abelian
black holes, which goes beyond linearized perturbations. Some elementary
entropy considerations are also presented...
