We present an in-depth first-principles study of the lattice dynamics of the tin sulphides SnS2, Pnma and π-cubic SnS and Sn2S3. An analysis of the harmonic phonon dispersion and vibrational density of states reveals phonon bandgaps between low- and high-frequency modes consisting of Sn and S motion, respectively, and evidences a bond-strength hierarchy in the low-dimensional SnS2, Pnma SnS and Sn2S3 crystals. We model and perform a complete characterisation of the infrared and Raman spectra, including temperature-dependent anharmonic linewidths calculated using many-body perturbation theory. We illustrate how vibrational spectroscopy could be used to identify and characterise phase impurities in tin sulphide samples. The spectral linewidths are used to model the thermal transport, and the calculations indicate that the low-dimensional Sn2S3 has a very low lattice thermal conductivity, potentially giving it superior performance to SnS as a candidate thermoelectric material.