It is demonstrated that well-known hydrodynamic instabilities can emerge spontaneously in microscopic particle systems. These phenomena are relatively difficult to obtain, since they require large particle ensembles and/or long simulation runs. This study uses a microscopic molecular-dynamics system with rare-gas-like atoms interacting via a Lennard-Jones potential. Because of the large number of particles, the system itself may be regarded as mesoscale, covering the space-time domain of micrometers and nanoseconds. In this study, the atomic parameters are not treated rigorously but, rather, are adjusted to observe their influence on the dynamics of the system.