We present a local stability analysis to investigate the effects of differential diffusion between momentum and density (quantified by the Schmidt number ) on the three-dimensional, short-wavelength instabilities in planar vortices with a uniform stable stratification along the vorticity axis. Assuming small diffusion in both momentum and density, but arbitrary values for , we present a detailed analytical/numerical analysis for three different classes of base flows: (i) an axisymmetric vortex, (ii) an elliptical vortex and (iii) the flow in the neighbourhood of a hyperbolic stagnation point. While a centrifugally stable axisymmetric vortex remains stable for any , it is shown that can have significant effects in a centrifugally unstable axisymmetric vortex: the range of unstable perturbations increases when is taken away from unity, with the extent of increase being larger for than for . Additionally, for 1$]]>, we report the possibility of oscillatory instability. In an elliptical vortex with a stable stratification, is shown to non-trivially influence the three different inviscid instabilities (subharmonic, fundamental and superharmonic) that have been previously reported, and also introduce a new branch of oscillatory instability that is not present at . The unstable parameter space for the subharmonic (instability IA) and fundamental (instability IB) inviscid instabilities are shown to be significantly increased for , respectively. Importantly, for sufficiently small and large and 1$]], respectively, the maximum growth rate for instabilities IA and IB occurs away from the inviscid limit. The new oscillatory instability (instability III) is shown to occur only for sufficiently small