The stability conditions for restricted closed shell Kohn-Sham density functional treatments are derived. In close analogy to the Hartree-Fock method we obtain the equivalent of singlet and triplet instabilities. The nonreal instability now becomes trivial and just expresses the aufbau principle. Demonstrative applications confirm the expected increased stability of density functional treatments as compared to Hartree-Fock. This is an improvement but the unphysical bifurcations do persist.