  Two-dimensional (2D) materials are known to possess emergent properties that
are not found in their bulk counterparts. Recent experiments have shown a
$\sqrt7 \times \sqrt3$ charge density wave (CDW) in monolayer 1T-VSe$_2$, in
contrast to the $4\times 4\times 3$ phase in bulk. Here, via first-principles
calculations, we show that multiple CDW phases compete in monolayer VSe$_2$,
the ground state of which can be tuned by charge doping and in-plane biaxial
strain. With doping, the $\sqrt7 \times \sqrt3$ CDW of the pristine VSe$_2$
transfers to a $3 \times \sqrt3$ and $4\times 4$ phase, the latter of which is
a projection of the bulk counterpart, at critical doping concentrations of
around 0.2 holes per formula unit and 0.25 electrons per formula unit,
respectively. The $4\times 4$ CDW phase can also be stabilized under
compressive strain. Although electron-phonon coupling is prevailing in the CDW
formation, we show that Fermi surface nesting is a good starting point to
explain most of these transitions in monolayer 1T-VSe$_2$. These results make
VSe$_2$ an appealing material for electronic devices based on controllable CDW
phase transitions.
