The tin sulfides represent a materials platform for earth-abundant semiconductor technologies. We present a first-principles study of the five known and proposed phases of SnS together with SnS2 and Sn2S3. Lattice-dynamics techniques are used to evaluate the dynamical stability and temperature-dependent thermodynamic free energy, and we also consider the effect of dispersion forces on the energetics. The recently identified π-cubic phase of SnS is found to be metastable with respect to the well-known orthorhombic Pnma/Cmcm equilibrium. The Cmcm phase is a low-lying saddle point between Pnma local minima on the potential-energy surface and is observed as an average structure at high temperatures. Bulk rocksalt and zincblende phases are found to be dynamically unstable, and we show that whereas rocksalt SnS can potentially be stabilized under a reduction of the lattice constant the hypothetical zincblende phase proposed in several previous studies is extremely unlikely to form. We also investigate the stability of Sn2S3 with respect to SnS and SnS2 and find that both dispersion forces and vibrational contributions to the free energy are required to explain its experimentally observed resistance to decomposition.