Two-dimensional layered materials with different anions on both faces of each layer, called Janus monolayers, have attracted significant interest due to their unique structural asymmetry perpendicular to the layer, which gives rise to an electric dipole moment that further diversifies the versatile properties of the transition-metal dichalcogenide materials class. However, the synthesis of a material such as MoSSe is prone to the introduction of various point defects, which could significantly modify the electric and optical properties as well as vibrational spectra. Using density functional theory, we provide an in-depth insight into the thermal stability of numerous point-vacancy and antisite defects in the Janus MoSSe monolayer. The structural changes are discussed in terms of the local strain induced by the modified atomic bonding around the defect sites. The electronic structure and linear optical response of Janus MoSSe monolayer with various point defects are studied, and possible fingerprints of electronic transitions due to defects are rationalized. First-principles calculations of phonons are carried out to spot the fingerprint of each point defect in the vibrational spectrum of the Janus MoSSe monolayer. Our systemic study will provide a broad picture of the roles the point defects could play in modifying and tuning the electronic and optical properties of two-dimensional Janus materials and thus help customizing them for certain applications.