Atomic layer-deposited ZrO(2) (zirconia) and HfO(2) (hafnia) films with various thicknesses, ranging from 112 to 660 nm, have been studied by Raman scattering spectroscopy. Spectral analysis of the excellent quality Raman data obtained by using freestanding edges of the films has unambiguously demonstrated that a metastable tetragonal t-ZrO(2) is coexisting with the stable monoclinic phase in zirconia films. Even though the Raman spectrum signal-to-noise ratio was high, only the monoclinic phase was positively identified from the observed spectral patterns of hafnia films. X-ray diffraction patterns are used to define the structure of metastable phases. Complementary Brillouin light scattering measurements of the freestanding edges are also employed in constraining elastic properties of the 405 nm HfO(2) thin film.