Diffusive instabilities of the Mullins-Sekerka type are one of the principal mechanisms through which microstructures form during solidification. In this study, we perform a linear stability analysis for the perturbation of a planar interface, where we derive analytical expressions to characterize the dispersion behavior in multi-component alloys under directional and isothermal solidification conditions. Subsequently, we confirm our calculations using phase-field simulations for different choices of the inter-diffusivity matrices. Thereafter, we highlight the characteristics of the dispersion curves upon change of the diffusivity matrix and the velocity. Finally, we also depict conditions for absolute stability of a planar interface under directional solidification conditions.