Prototype inertial confinement fusion targets for the proposed National Ignition Facility are metallic or plastic spherical shells (2 mm o.d., (approximately equal to)150-mm thickness) with an (approximately equal to)80-mm-thick layer of solid deuterium--tritium (50/50 mixture) deposited on the inner surface. Ignition will occur only if the D--T fuel layer meets strict sphericity and surface roughness criteria (typically (approximately equal to)1 mm). Symmetric layering of solid D--T occurs due to the phenomenon of ``beta layering'' in which tritium-induced self-heating drives the redistribution of material. In contrast to the optical techniques usually employed, this work discusses methods in which resonant ultrasound spectroscopy (RUS) and related techniques can be used to help determine the uniformity of the fuel layer inside opaque targets. A tetrahedral array of pinducers in a cryogenic apparatus is used to both mount and probe the sample. Preliminary efforts have focused on the characterization of solid spheres and both aluminum and beryllium shells. Sufficiently high Q's (10[sup 3]-10[sup 5]) necessary for detailed work are readily obtained. Studies involving deuterium-filled shells include the observation of liquid condensation, triple point measurement, and the effects of thermally induced redistribution.