  We consider the minimally coupled 3-form field which has been considered as a
candidate to realize inflation. We have studied the conditions to avoid ghosts
and Laplacian instabilities and found that some classes of potentials, e.g. the
Mexican-hat one, will in general be unstable. We then propose other classes of
potentials which are instead free from any instability, drive a long-enough
slow-roll regime followed by an oscillatory epoch, and as a consequence, can
provide successful inflation. Finally, we also provide stable potentials which
lead to a small enough propagation speed for the scalar perturbations, giving a
possibility for these models to produce non-Gaussianities.
