  Based on the secular spin evolution of black holes, neutron stars,
gravastars, or boson stars in precessing compact binaries on eccentric orbit,
we identify the aligned and more generic coplanar configurations of the spins
and orbital angular momentum as fix points. Through a dynamical system
analysis, we investigate their linear stability as function of the mass
quadrupole parameter. Marginal stability holds for the binary configurations
with both spins antialigned to the orbital angular momentum, for both spins
aligned to the orbital angular momentum (with the exception of certain
quadrupolar parameter ranges of neutron stars and boson stars), and for the
extremal mass ratio. For equal masses, the configurations of one of the spins
aligned and the other antialigned is stable for gravastar binaries, for neutron
star binaries in the high quadrupolar parameter range, and for boson star
binaries. For some unequal mass gravastar binaries, black hole binaries or
neutron star binaries, a transition from stability to instability can occur
during the inspiral, when one of the spins is aligned, while the other is
antialigned to the orbital angular momentum. We discover a transitional
instability regime during the inspiral of certain gravastar, neutron star, or
boson star binaries with opposing spins. For coplanar configurations we find
instabilities only for the gravastar - gravastar, boson star - boson star and
black hole - boson star binaries. For a given spin configuration, marginal
stability strongly depends on the value of the quadrupolar parameters. The
stability region is larger for neutron star binaries than for black hole
binaries, while the mixed systems have a restricted stability parameter region.
