A supersolid is an elastic solid which is partly superfluid. Supersolidity which has been experi- mentally observed in 4He in 2004, seems paradoxical and is still controversial. Below 200 mK where supersolidity occurs, the shear modulus increases. Both properties are disorder dependent and the transition temperature depends of 3He impurity concentration even for concentration as low as a ppb. In this thesis, we have studied 4He single crystals mechanical properties, in the limit of extreme quality and purity. We have built acoustic cavities with piezoelectric transducers, allowing to measure acoustic resonances in 4He single crystals. In which we measure optically the crystalline orientation. Our results show that dislocations are responsible for the elastic coefficients anomaly. They show that the pinning of dislocations by 3He impurities at low temperature explains the rigidity change of crystals. The change in the elastic coefficient can be as large as ∼86 % and must involve the motion of dislocation lines along macroscopic distances at relatively large velocity. Also we have measured the moment of inertia of 4He crystals in a transparent torsional oscillator. Single crystals show a rotation anomaly which is more explicable by an elastic anomaly than in terms of supersolid transition. Our results illustrates principally the remarkable quantum plasticity of 4He crystals.