  The stability problem of non-Abelian monopoles with respect to
"Brandt-Neri-Coleman type" variations reduces to that of a pure gauge theory on
the two-sphere. Each topological sector admits exactly one stable monopole
charge, and each unstable monopole admits $2\sum (2|q|-1)$ negative modes,
where the sum goes over the negative eigenvalues $q$ of an operator related to
the non-Abelian charge $Q$ of Goddard, Nuyts and Olive. An explicit
construction for the [up-to-conjugation] unique stable charge, as well as the
negative modes of the Hessian at any other charge is given. The relation to
loops in the residual group is explained. From the global point of view, the
instability is associated with energy-reducing two-spheres, which, consistently
with the Morse theory, generate the homology of the configurations space, and
whose tangent vectors at a critical point are negative modes. Our spheres might
indicate possible decay routes of an unstable monopole as a cascade into lower
lying critical points.
