Is there a way to predict the melting point of a solid from phonon dispersion curves, measured experimentally or via DFT?
There is a rule-of-thumb known as the "Lindemann criterion", which says that the melting point should occur when the root mean square thermal motion of a single atom is about 10% of the interatomic separation. This is not quantitatively accurate (10% is basically an arbitrary number) but a simple estimate which has a rough correlation to reality.
Using the dispersion relation one can estimate this thermal motion in the harmonic approximation, and thus make some very rough estimate of the melting temperature. Note that any calculation made using the dispersion relation must be approximate, since it only includes harmonic order in the potential, and thermal fluctuations in the lattice should be reasonably large for the system to melt (the entire idea of this criterion!)