Experiments were conducted in a rectangular supercritical carbon dioxide (SCCO2) natural-circulation loop at Argonne National Laboratory (ANL) in order to verify the stability margin as suggested by some previous investigators. Although a one-dimensional transient computational model developed at University of Wisconsin, Madison, predicted the development of instabilities for the SCCO2 loop, which had good agreement with some previous work, the experiments conducted at the ANL SCCO2 loop exhibited stable behavior under similar conditions. In order to bridge the gap between the numerical predictions and experimental results by distinguishing between the numerical effects and physical effects, a linear stability approach is adopted in the present study. The linear stability analysis has been conducted for three model natural-circulation loop geometries employing water or carbon dioxide as the working fluid. The results for the supercritical water loops displayed flow stability for a more accurate equation of state (EOS); however, the analysis indicated the presence of instabilities for a less accurate EOS. Furthermore, this analysis still predicts the presence of instabilities for the SCCO2 loop similar to our transient numerical predictions. We additionally note that the stability margin for both water loops and the SCCO2 loop does not correspond with proposed stability criteria from a previous analysis. These two final points suggest the phenomenon is a more complex function of both fluid properties and loop geometry.