A rock held above the ground has potential energy. As the rock falls, this potential energy is converted to kinetic energy. Finally, the rock hits the ground and stays there. What has happened to the energy?

By the law of conservation of mechanical energy, in the absence of any drag, when the rock falls towards the ground, the potential energy gets converted to kinetic energy. Once the rock hits the ground, it does not move. This is because, the energy is dissipated as sound or heat (to the environment). The total energy remains conserved. Only the form of energy changes.