  First-principles calculations, within the framework of density functional
theory, have been performed on the well-studied 2H and the less explored
1T$^{\prime}$ phase of single-layer MoS$_{2}$. We have addressed the
strain-induced tunability of the electronic and phononic properties of both
phases, and compared their stability against the applied strain. By considering
a large number of strain profiles for both tensile and compressive stress, we
have found that the electronic properties of both 2H and 1T$^{\prime}$ phases
are sensitive to the direction of the applied strain and can be tuned in a
controlled way. For the 2H phase, in most cases, a direct to indirect band gap
transition at lower strain and a semiconductor to metal transition at higher
strain is observed. The applied strain destroys the semimetallic nature of the
1T$^{\prime}$ phase via the overlapping of the bulk states with the
topologically protected edge states. Significant strain-induced changes in the
phononic properties, in the frequency of the phonon branches, as well as in the
nature of the dispersion curves, are observed. A systematic change in the
frequency of the optical phonon modes at the zone centre is seen for both
phases. With increasing strain, the out-of-plane acoustic mode (ZA) turns
imaginary, indicating a possibility of phase transition or instability of the
crystal structure. The 2H phase appears to withstand a larger amount of strain
and therefore possesses better stability compared to the 1T$^{\prime}$ phase
since the imaginary branch starts to appear at much lower values of strain in
the latter case. We highlight the significance of strain engineering in tuning
the electronic and phononic properties and the safe limit of the strain
application in different polymorphs of single-layer MoS$_{2}$.
