  We study the stability of three analytical solutions of the Einstein's field
equations for spheres of fluid. These solutions are suitable to describe
compact objects including white dwarfs, neutron stars and supermassive stars
and they have been extensively employed in the literature. We re-examine the
range of stability of the Tolman VII solution, we focus on the stability of the
Buchdahl solution which is under contradiction in the literature and we examine
the stability of the Nariai IV solution. We found that all the mentioned
solutions are stable in an extensive range of the compactness parameter. We
also concentrate on the effect of the adiabatic index on the instability
condition. We found that the critical adiabatic index, depends linearly on the
ratio of central pressure over central energy density $P_c/{\cal E}_c$, up to
high values of the compactness. Finally, we examine the possibility to impose
constraints, via the adiabatic index, on realistic equations of state in order
to ensure stable configurations of compact objects.
