We consider three left-invariant Lorentzian structures on the Heisenberg group, which differ by the position of the future light cone relative to the commutator subalgebra of the Heisenberg algebra. For these structures, we investigate the reachable set, the existence of Lorentzian length maximizers, the parametrization of extremal trajectories, and the geometry of wave fronts. The local and global optimality of extremal trajectories is studied using both analytical and numerical methods.