We explore spin-preserving, singlet stability of restricted Hartree-Fock (RHF) solutions for a number of closed-shell, homonuclear diatomics in the entire relevant range of internuclear separations. In the presence of such instabilities we explore the implied broken-symmetry (bs) solutions and check their stability. We also address the occurrence of vanishing roots rendered by the stability problem in the case of bs solutions. The RHF bs solutions arise primarily due to the symmetry breaking of the relevant, mostly frontier, molecular orbitals, which approach atomic-type orbitals in the dissociation limit. The resulting bs RHF solutions yield more realistic potential energy curves (PECs) than do the symmetry adapted (sa) solutions. These PECs are shown to be very similar to those rendered by the density functional theory (DFT). Moreover, the sa DFT solutions are found to be stable in a much wider range of internuclear separations than are the RHF solutions, and their bs analogs differ very little from the sa ones. Finally, we examine a possible usefulness of bs RHF solutions in post-HF correlated approaches to the many-electron problem, specifically in the limited configuration interaction and coupled-cluster methods.