Recently, Kim and Chan (Science 305:1941, 2004; Phys. Rev. Lett. 97:115302, 2006) have reported an anomalous decoupling transition of solid 4He in a torsional oscillator measurement, and interpret their results as evidence for non-classical rotational inertia and a possible supersolid phase of 4He. The detailed nature and properties of such a “supersolid” state in 4He are still far from being clear, although there are clues from experiments involving 3He impurities, different sample cell geometries, annealing effects and grain boundary flow. Defects produced during crystal growth or deformation (e.g. dislocations) may affect supersolidity, or even produce it, and they are expected to have significant impact on the elastic properties of the solid. The supersolid fraction could also decouple from the lattice and produce a decrease in the transverse sound speed. We have begun the experiments in this laboratory to study such effects, measuring the velocity and attenuation of transverse ultrasound at 10 MHz in 4He single crystals grown at constant pressure.