  Chemically and mechanically exfoliated MoS$_2$ single-layer samples have
substantially different properties. While mechanically exfoliated single-layers
are mono-phase (1H polytype with Mo in trigonal prismatic coordination), the
chemically exfoliated samples show coexistence of three different phases, 1H,
1T (Mo in octahedral coordination) and 1T$^{'}$ (a distorted $2\times 1$
1T-superstructure). By using first-principles calculations, we investigate the
energetics and the dynamical stability of the three phases. We show that the 1H
phase is the most stable one, while the metallic 1T phase, strongly unstable,
undergoes a phase transition towards a metastable and insulating 1T$^{'}$
structure composed of separated zig-zag chains. We calculate electronic
structure, phonon dispersion, Raman frequencies and intensities for the
1T$^{'}$ structure. We provide a microscopical description of the J$_1$, J$_2$
and J$_3$ Raman features first detected more then $20$ years ago, but
unexplained up to now. Finally, we show that H adsorbates, that are naturally
present at the end of the chemical exfoliation process, stabilize the
1T$^{\prime}$ over the 1H one.
