Active filaments, such as kinesin propelled microtubules in gliding assay experiments, give rise to a plethora of active phases. In order to better understand which features of these phases are emergent and which exist at even the single filament level, we investigate the dynamics of individual active elastic filaments with chiral self-propulsion. To this end, we study the fully general time evolution of an overdamped plane curve and derive equations for the evolution of the curve’s shape and orientational characteristics. Applying this formalism to the specific case of an active elastic filament with chiral self-propulsion, we determine that sufficiently flexible filaments can exhibit stationary states with shape multi-stability which in turn gives rise to rotational dynamics. Further, the time-dependent evolution towards such steady states is highly nontrivial with both wave-like and diffusive characteristics available depending on the elastic properties of the system.