Nowadays, S-nitrosothiols (RSNOs) represent a promising class of nitric oxide (NO) donors that could be successfully used as drugs to compensate the decrease of NO production that usually arises in conjunction with cardiovascular diseases. Nevertheless, notwithstanding their pharmacological interest, the structure–stability relationship in RSNOs is still unclear, and this issue, together with the mechanism of NO donation in the physiological medium, deserves further investigation. As a first step forward in this direction, in this paper, the overall stability and structural preference of two pharmacologically relevant S-nitrosothiol molecules were studied in detail by means of computational strategies. In particular, performing calculations in implicit solvent (water) on the S-nitroso-N-acetylpenicillamine and the S-nitroso-N-acetylcysteine and analyzing the noncovalent interactions networks of their most stable conformers, we observed that the structure and the stability of these molecules can be directly related to the formation of stabilizing hydrogen-bond and chalcogen–chalcogen intramolecular interactions. The obtained results represent the starting point for further investigations to be conducted also on larger RSNOs to shed further light on the role played by intra- and intermolecular interactions and by solvation effects in stabilizing this class of molecules. The obtained insights will be hopefully helpful to design new RSNO-based drugs characterized by an enhanced pharmacological potency.