We present an experimental study of the quasi-elastic Raman scattering (QES) of plane-wave and twisted light by liquid crystals. Depending on their temperature, these crystals can exhibit isotropic, nematic and chiral nematic phases. The question is addressed of how the phase of a crystal and the state of incident light can affect the quasi-elastic energy spectra of the scattered radiation, whose shape is usually described by the combination of Lorentzian and Gaussian components. Special attention is paid to the chiral phase, for which the Raman QES spectrum is dominated by a Lorentzian with reduced linewidth, pointing to diminished disorder and configurational entropy. Moreover, this phase is also known for a regime of iridescence (selective backscattering) which arises when the wavelength of incident light becomes comparable with the chiral pitch length. Detailed measurements, performed in this resonant regime and by employing twisted light, carrying various projections of the orbital angular momentum (OAM), have indicated a low-energy scattering surplus depending on OAM. We argue that this observation might indicate a transfer of angular momentum between light and liquid crystal.