We present Raman and terahertz absorbance spectra of methylammonium lead halide single crystals (MAPbX3, X = I, Br, Cl) at temperatures between 80 and 370 K. These results show good agreement with density-functional-theory phonon calculations. Comparison of experimental spectra and calculated vibrational modes enables confident assignment of most of the vibrational features between 50 and 3500 cm-1. Reorientation of the methylammonium cations, unlocked in their cavities at the orthorhombic-to-tetragonal phase transition, plays a key role in shaping the vibrational spectra of the different compounds. Calculations show that these dynamic effects split Raman peaks and create more structure than predicted from the independent harmonic modes. This explains the presence of extra peaks in the experimental spectra that have been a source of confusion in earlier studies. We discuss singular features, in particular the torsional vibration of the C-N axis, which is the only molecular mode that is strongly influenced by the size of the lattice. From analysis of the spectral linewidths, we find that MAPbI3 shows exceptionally short phonon lifetimes, which can be linked to low lattice thermal conductivity. We show that optical rather than acoustic phonon scattering is likely to prevail at room temperature in these materials.