A local stability analysis of idealized models of stratified vortices that appear in various engineering and geophysical settings is presented. Specifically, detailed short-wavelength stability analysis is performed for three different classes of base flows: (a) an axisymmetric barotropic vortex with an out-of-plane stable stratification, (b) an axisymmetric baroclinic vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in thermal wind balance with the out-of-plane momentum gradient, and (c) an elliptical barotropic vortex with an out-of-plane stable stratification. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on Schmidt number Sc (ratio between momentum and mass diffusivities) is studied. The role of non-unity Sc in affecting the inviscid instabilities, and in introducing new instabilities is discussed in detail for a wide range of base flow and perturbation parameters.