Imaginary frequency instantaneous normal modes (INM) of a number of ionic melts are examined in the light of recent proposals which connect them to diffusion and to the strong/liquid character of the fluid. Separate INM density-of-states may be found for each ionic species. A linear relationship between the temperature dependence of the fraction of INMs which are unstable (f(u)) and that of the diffusion coefficient of each species has been found. The shapes of the single-mode potential energy curves of the fluid along INM eigenvectors are examined. The results do not support a general significance for "double-well" modes as the only ones involved in diffusion. An alternative explanation of how single or double well features develop on the single-mode potential curves is proposed. The shape of the unstable INM density-of-states reflects the character of the ionic motion: a crossover from \omega\exp(-B omega(2)) to \omega\exp(-B omega(4)) occurs as the ionic motion becomes increasingly hindered. Both behaviors, which have been reported previously for the Lennard-Jones atomic fluid at normal liquid and supercooled conditions, respectively, can occur simultaneously in a single melt for each ionic species.