  For monopoles with nonvanishing Higgs potential it is shown that with respect
to "Brandt-Neri-Coleman type" variations (a) the stability problem reduces to
that of a pure gauge theory on the two-sphere (b) each topological sector
admits one, and only one, stable monopole charge, and (c) each unstable
monopole admits $2\sum_{q<0} (2|q|-1)$ negative modes, where the sum goes over
all negative eigenvalues $q$ of the non-Abelian charge $Q$. An explicit
construction for (i) the unique stable charge (ii) the negative modes and (iii)
the spectrum of the Hessian, on the 2-sphere, is then given. The relation to
loops in the residual group is explained. The negative modes are tangent to
suitable energy-reducing two-spheres. The general theory is illustrated for the
little groups U(2), U(3), SU(3)/Z_3 and O(5).
