  We present a polarization-resolved, high-resolution Raman scattering study of
the three consecutive charge density wave (CDW) regimes in $1T$-TaS$_2$ single
crystals, supported by \textit{ab initio} calculations. Our analysis of the
spectra within the low-temperature commensurate (C-CDW) regime shows
$\mathrm{P3}$ symmetry of the system, thus excluding the previously proposed
triclinic stacking of the "star-of-David" structure, and promoting trigonal or
hexagonal stacking instead. The spectra of the high-temperature incommensurate
(IC-CDW) phase directly project the phonon density of states due to the
breaking of the translational invariance, supplemented by sizeable
electron-phonon coupling. Between 200 and 352\,K, our Raman spectra show
contributions from both the IC-CDW and the C-CDW phase, indicating their
coexistence in the so-called nearly-commensurate (NC-CDW) phase. The
temperature-dependence of the symmetry-resolved Raman conductivity indicates
the stepwise reduction of the density of states in the CDW phases, followed by
a Mott transition within the C-CDW phase. We determine the size of the Mott gap
to be $\Omega_{\rm gap}\approx 170-190$ meV, and track its temperature
dependence.
