The longitudinal susceptibility of the low-temperature Heisenberg ferromagnet is treated in random-phase approximation (RPA) in order to discuss whether a zero magnon (spin-density mode in the collisionless regime) can exist. In the unrenormalized RPA, this mode is found at large wave vectors in systems which have an exchange anisotropy which enhances the repulsion between spin-density fluctuations. It is shown, however, that (with the possible exception of certain two-dimensional systems) spin-wave damping suppresses this mode. A similar approach is applied to the low-temperature Heisenberg antiferromagnet. For systems with non-negligible anisotropy, it is shown that, with the possible exception of certain two-dimensional systems, spin-wave damping suppresses the zero-magnon mode. An argument is then given which suggests that the mode does not exist in the isotropic antiferromagnet either.