Due to the ever increasing power and cooling requirements of large-scale computing and data facilities, there is a worldwide search for low-power alternatives to CMOS. One approach under consideration is superconducting computing based on single-flux-quantum logic. Unfortunately, there is not yet a low-power, high-density superconducting memory technology that is fully compatible with superconducting logic. We are working toward developing cryogenic memory based on Josephson junctions that contain two or more ferromagnetic (F) layers. Such junctions have been demonstrated to be programmable by changing the relative direction of the F layer magnetizations. There are at least two different types of such junctions -- those that carry the innate spin-singlet supercurrent associated with the conventional superconducting electrodes, and those that convert spin-singlet to spin-triplet supercurrent in the middle of the device. In this paper we compare the performance and requirements of the two kinds of junctions. Whereas the spin-singlet junctions need only two ferromagnetic layers to function, the spin-triplet junctions require at least three. In the devices demonstrated to date, the spin-singlet junctions have considerably larger critical current densities than the spin-triplet devices. On the other hand, the spin-triplet devices have less stringent constraints on the thicknesses of the F layers, which might be beneficial in large-scale manufacturing.