  Self-consistent mean field methods based on phenomenological Skyrme effective
interactions are known to exhibit spurious spin and spin-isospin instabilities
both at zero and finite temperatures when applied to homogeneous nuclear matter
at the densities encountered in neutron stars and in supernova cores. The
origin of these instabilities is revisited in the framework of the nuclear
energy density functional theory and a simple prescription is proposed to
remove them. The stability of several Skyrme parametrizations is reexamined.
