We have studied the concentration dependence of the Raman spectra of heteroepitaxial $({\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{Sr}}_{x}){\mathrm{TiO}}_{3}$ (BST) $(x=0,$ 0.15, 0.30, and 0.45) thin films deposited by rf sputtering on (001) MgO substrates. It is found that substitution of Ba by Sr leads to the distortions of the Ti-O bonds and causes the appearance of new bands in the Raman spectra near the bands corresponding to the Ti-O vibrations. The Sr incorporation induces relatively large shifts of the low-frequency Raman peaks attributed to ${E}_{\mathrm{TO}}$ and ${A}_{1\mathrm{TO}}$ components of the soft mode. The ${E}_{\mathrm{TO}}$ soft mode in BST films becomes underdamped for $xg~0.3$ and is very sensitive to the two-dimensional compressive stresses.