The paper collects and discusses findings emerging from the analysis of systems operating with fluids at supercritical pressure, with reference to flow stability. In particular, the influence of heating structures and numerical diffusion on the predicted dynamic behavior is highlighted, clarifying that results obtained paying little attention to the presence of these effects should be reconsidered for a better realistic prediction of stability characteristics. Examples of applications in which truncation error and the presence of heating structures play an important role are reported, in order to warn about a tendency to underestimate these effects on the basis of the knowledge of similar phenomena (e.g., in two-phase flow) or system configurations in which they might play a lesser role. The use of a computational fluid dynamics (CFD) code in the analysis of a simple single-tube stability problem shows that models more complex than the usual one-dimensional (1D) ones also show similar effects. The results obtained by 1D numerical tools developed for the analysis of natural circulation with supercritical pressure fluids, equipped with the capability to simulate linear and nonlinear stability with first- and second-order explicit schemes, are then reported. The discussion of the eigenvalues and the eigenvectors calculated for an existing natural circulation loop and a single channel highlight interesting aspects that can be helpful in understanding the results of stability analyses. The CFD code analysis adds additional aspects of interest for the discussion.