Harmonic and anharmonic vibrational frequencies, as well as infrared and Raman intensities, are calculated for MgSO4.nH(2)O (n=1-3). Electronic structure theory at the second order Moller-Plesset perturbation theory level with a triple-zeta+polarization basis set is used to determine the geometry, energetics, charge distributions, and vibrational spectra of pure and hydrated MgSO4 salts. The direct vibrational self-consistent field method and its correlation corrected extension are used to determine anharmonic corrections to vibrational frequencies and infrared intensities for the pure MgSO4 and its complex with one water molecule. Very significant differences are found between vibrational spectra of water molecules in complexes with MgSO4 and pure water. Some of the O-H stretching frequencies are shifted to the red very significantly (by up to 2000 cm(-1)) upon complexation with magnesium sulfate. They should be observed between 1700 and 3000 cm(-1), in a region very different from the corresponding O-H stretch frequency region of pure water (3700-3800 cm(-1)). In addition, the SO2 stretching vibrations are found at lower frequency regions than the water vibrations. They can serve as unique identifiers for the presence of sulfate salts. The predicted infrared and Raman spectra should be of valuable help in the design of future missions and analysis of observed data from the ice surface of Jupiter's moon Europa that possibly contains hydrated MgSO4 salts.