Ultrasonics have been an incisive probe of internal interfaces in a wide variety of systems ranging from stars to solids. For thin-film structures, however, ultrasound is largely ineffective because the signal is dominated by the substrate. Using confocal Raman spectromicroscopy, we show that multiple reflection of sound waves at internal interfaces of a metal-oxide superlattice generates standing waves that are insensitive to the substrate. Such modes had previously been observed only in high-quality superlattices of elemental semiconductors, and their observation in complex metal-oxide heterostructures is testimony to recent progress in this field. We use the high spatial resolution of the Raman microscope to demonstrate the high sensitivity of the mode frequency to atomic-scale thickness variations of the superlattice. Spectroscopy of acoustic standing waves can hence serve as a powerful characterization tool of thin-film structures. In analogy to ultrasound spectroscopy of bulk solids, lineshape analysis of these modes has the potential to yield detailed information about the internal structure of the interfaces as well as the coupling of sound waves to the low-frequency spin, charge, and orbital dynamics in metal-oxide superlattices.