Generally, in D=1,2 , formation of charge density waves is accompanied by the Kohn anomaly , i.e. the vanishing (in D=1) or a dip (in D=2) of a phonon mode in a phonon dispersion spectrum.

The formation of CDW and the Kohn anomaly are complementary phenomena of a metal-insulator transition due to interaction of electron and phonon system either in electron or phonon picture.

Dear Prof. Dr Aga Shahee

Your question is interesting & quite relevant, henceforth I will try to answer partially your question in a couple of posts in your thread:

A sound velocity anomaly can be directly calculated by means of Ehrenfest relations for 2nd order phase transitions-PT. For instance a jump in sound velocity can be calculated at the superconducting transition temperature Tc. This is an anomaly that have been observed in some materials. See [1] for example to have a better idea, but I can refer to some specific literature if you wish.

[1] https://www.worldscientific.com/doi/pdf/10.1142/9789813274181_0001

This is part of the question, but what I cannot do now is to relate the jump to an structural phase transition which I guess is a 1st order PT [2] quoting & is associated with a dynamical instability by the soft mode. According to some authors [3] a physical explanation will be "...the soft mode happens when cooling a material from a temperature above Tc, a normal mode of vibration of the crystal decreases to 0 freq when the crystal becomes unstable & distorts to a new structure..."

[2] https://arxiv.org/ftp/arxiv/papers/1102/1102.1085.pdf

[3] https://arxiv.org/ftp/cond-mat/papers/0605/0605489.pdf

Finally, respect to lattice degrees of freedom showing divergence, I guess this is what happens to 1D systems (according to Peierls & Landau [4]), isn´t it?

[4] https://en.wikipedia.org/wiki/Peierls_transition