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MoSi2N4 is a two-dimensional ternary nitride semiconductor that has attracted attention for its excellent mechanical and thermal properties. Theoretical studies predict that zigzag edges of this material can host magnetic edge states and Dirac fermions, but the stability of such edges has not been examined. Here, we present a density functional theory study of the electronic and thermodynamic properties of MoSi2N4 edges. We develop a (partial) ternary phase diagram that identifies a region of chemical potentials within which MoSi2N4 is stable over competing elemental or binary phases. Based on this phase diagram, we determine the thermodynamic stability of several armchair and zigzag edges and elucidate their electronic structures. Bare zigzag edges, predicted to host exotic electronic states, are found to be substantially higher in energy than armchair edges and, thus, unlikely to occur in practice. However, with hydrogen passivation, these zigzag edges can be stabilized relative to their armchair counterparts while retaining metallicity and magnetic order. Our analysis provides a solid thermodynamic basis for further exploration of MoSi2N4 in nanoscale electronics and spintronics.