  We give an account of the various changes in the stability character in the
five types of Riemann ellipsoids by establishing the occurrence of different
quasi-periodic Hamiltonian bifurcations. Suitable symplectic changes of
coordinates, that is, linear and non-linear normal form transformations are
performed, leading to the characterisation of the bifurcations responsible of
the stability changes. Specifically we find three types of bifurcations,
namely, Hamiltonian pitchfork, saddle-centre and Hamiltonian-Hopf in the
four-degree-of-freedom Hamiltonian system resulting after reducing out the
symmetries of the problem. The approach is mainly analytical up to a point
where non-degeneracy conditions have to be checked numerically. We also deal
with the regimes in the parametric plane where Liapunov stability of the
ellipsoids is accomplished. This strong stability behaviour occurs only in two
of the five types of ellipsoids, at least deductible only from a linear
analysis.
