The anapole moment of a massless charged scalar particle is studied in a model-independent fashion, using the effective Lagrangian technique, as well as radiatively within the context of scalar quantum electrodynamics (SQED). It is shown that this gauge structure is characterized by a non-renormalizable interaction, which is radiatively generated at one-loop. It is found that the resulting anapole moment for off-shell particles, though free of ultraviolet divergences, is gauge dependent and thus it is not a physical observable. We also study some of its kinematical limits. In particular, it is shown that its value comes out to be zero when the photon is on-shell and the momenta squared of the incoming and outgoing scalars are equal. It is a stronger statement than it being zero for all particles being on-shell which is required by the current conservation.