Two-dimensional (2D) sheets of transition metal phosphorus trichalcogenides (TMPTs) offer unique magnetic and optical properties that can complement those found in other 2D materials. Insights into the structure and properties of these materials can be obtained by a juxtaposition of the experimental and calculated Raman spectra, but there is very little theoretical knowledge of the vibrational properties of TMPTs. Using first-principles calculations, we study mechanical and vibrational properties of a large set of monolayer TMPTs. From the phonon dispersion curves, we assess the dynamical stabilities and general trends on the atomic character of the vibrational modes. We determine Raman active modes from group theory, calculate Raman intensities, and analyze them with the help of the corresponding atomic displacements. We evaluate how the mode frequencies shift in response to a biaxial strain. We also determine elastic properties, which show that these systems are softer than many other layered materials. In addition to shedding light on the general features of vibrational properties of these materials, our results should also prove useful for interpreting experimental Raman spectra.