They aren't neglected.  If the electron is free, there aren't any corrections, if the electron does interact, then they can be computed by standard methods. 

Dr. Pankaj Bhalla, inelastic scattering of phonons can occur for example due to Umklapp processes.

See Wikipedia commons for the definition:

https://en.wikipedia.org/wiki/Umklapp_scattering

Concerning the Migdal theorem, I have no seen too much experimental data, although, there is a significant amount of theoretical works, see for example:

Article Migdal's theorem and electron-phonon vertex corrections in D...

Usually, fitting ultrasound attenuation data in superconductors there is no need for the vertex corrections.