The linearized governing equations for an ideal fluid are solved numerically for the stability of free capillary surfaces in rectangular containers against unfavorable disturbances (accelerations, i.e. Rayleigh-Taylor instability). The preliminary results are expressed graphically in terms of a critical Bond number as a function of system contact angle. A critical wetting phenomena in the corners is shown to significantly alter the region of stability for such containers when contrast to simpler geometries such as the circular cylinder or the infinite rectangular slot. Such computational results provide additional constraints for the design of fluids systems for space-based applications.