Motivated by observations of interleaving in the equatorial Pacific, we consider the linear stability of a basic state on an equatorial β-plane which is susceptible to both double-diffusive interleaving, driven by a meridional salinity gradient, and inertial instability driven by meridional shear. In a parameter regime compatible with the observations strong interaction can occur between the two processes, indicating that the stability of the system is dependent on the meridional gradients of both salinity and zonal velocity. Meridional shear is found to enhance the interleaving motion even for values of shear well below the cutoff for inertial instability. In the presence of diffusion inertial instability can also be excited by vertical shear, but only if the shear is comparable to the buoyancy frequency. When double-diffusive driving is weak relative to inertial driving the growth can be oscillatory, in which case the mechanism for instability is viscous–diffusive. In this case interleaving layers can slope downwards towards the fresh side of the front in the fingering regime, inhibiting their own growth.