This article considers two different problems in trajectory tracking control for linear systems. First, if the control is not unique, which is most input energy efficient. Second, if exact tracking is infeasible, which control performs most accurately. These are typical challenges for overactuated systems and for underactuated systems, respectively. We formulate both goals as optimal output regulation problems. Then, we contribute two new sets of regulator equations to output regulation theory that provide the desired solutions. A thorough study indicates solvability and uniqueness under weak assumptions. For example, we can always determine the solution of the classical regulator equations that is most input energy efficient. This is of great value if there are infinitely many solutions. We derive our results by a linear quadratic tracking approach and establish a useful link to output regulation theory.