Alexander and McTague [Phys.Rev. Lett. 41, 702 (1978)] argued that if there is a spinodal point associated with the liquid-solid transition in a fluid of spherically symmetric particles, the bcc phase will be uniquely favored as the only accessible symmetry breaking structure that forms a regular three-dimensional lattice. By reconsidering their analysis in the framework of density-functional theory, we show that at a liquid-solid spinodal in fact many other solid stuctures also are simultaneously accessible, among them the fcc structure. Nevertheless, the bcc structure is still shown to be special, as, independent of the details of the interaction, the free energy of the unstable bcc phase close to the spinodal is always lower than that of the other solidlike structures. We illustrate our general results by explicit calculations on a toy model, the "Onsager solid." This simple model also indicates that the ultimately stable crystal phase, which, as usual for sufficiently steep repulsive forces, turns out to be fcc, is dictated by properties of the free energy that cannot be obtained perturbatively starting from the spinodal point.