The equilibrium shape of a drop in contact with solid surfaces can undergo continuous or discontinuous transitions upon changes in either drop volume or surface energies. In many instances, such transitions involve the motion of the three-phase contact line and are thus sensitive to contact angle hysteresis. Using a combination of electrowetting-based experiments and numerical calculations, we demonstrate for a generic sphere-plate confinement geometry how contact angle hysteresis affects the mechanical stability of competing axisymmetric and nonaxisymmetric drop conformations and qualitatively changes the character of transitions between them.