Epitaxial thin films of lithium niobate with a thickness of 160 nm, oriented along the crystallographic c axis, were grown by direct liquid injection chemical vapor deposition on c-sapphire substrates. Different families of very high-frequency surface acoustic waves with general polarization exist in such piezoelectric films on high-velocity substrates. Surface Brillouin light scattering measurements, complemented with fast finite element analysis of wave dispersion, demonstrate Rayleigh, leaky shear, and leaky longitudinal surface waves, excited at frequencies between 10 and 30 GHz. The Brillouin technique reveals dispersion and anisotropy of propagation without the implementation of high-frequency surface acoustic wave transducers.