Janus structures with the breaking of out-of-plane mirror symmetry have gained intensive attention. Here, on the basis of first principles, we not only investigate the recently discovered Janus 2H-MoSH monolayer [ACS Nano 15, 20319 (2021)], but also report the global minimum structure of a Janus 1T-MoSH monolayer. Meanwhile, 2H-MoSH monolayer can easily transform into the 1T phase, overcoming a relatively small barrier of 0.13 eV. Intriguingly, Janus 1T-MoSH is a charge-density wave (CDW) material and its CDW order can be regulated via external strains. When Janus 1T-MoSH is subjected to 3% compressive strain, its CDW is completely suppressed and a superconducting state with a superconducting transition temperature $\mathit{T}{}_{c}$ of 25.15 K emerges. Moreover, Janus 2H-MoSH monolayer is an intrinsic superconductor with a $\mathit{T}{}_{c}$ of $\ensuremath{\sim}26.81$ K and its $\mathit{T}{}_{c}$ can be enhanced up to 36.69 K under a 1% tensile strain. Our work provides more in-depth insights for superconducting and CDW in low-dimensional transition metal dichalcogenides and related Janus systems.