This paper studies the phenomenon of strain bifurcation and localization in J2 plasticity under plane stress and plane strain conditions. Necessary conditions for the outcome of bifurcation, localization, stress boundedness and decohesion are analytically established. It is shown that the explicit consideration of these conditions allows for the determination of localization angles in certain situations of interest that can be used to conduct benchmark tests on finite element formulations. The relative merits of irreducible, (stabilized) mixed and (displacement and/or strain) enhanced formulations are discussed. Numerical examples show that the mixed displacement/pressure formulation is to be preferred to the standard irreducible schemes in order to predict correct failure mechanisms with localized patterns of plastic deformation. Mixed elements are shown to be practically free from mesh directional bias dependence.