Supercritical fluids (SCFs) are known to exhibit anomalous behavior in their thermophysical properties such as diverging compressibility and vanishing thermal diffusivity on approaching the critical point. This behavior leads to a strong thermomechanical coupling when SCFs are subjected to simultaneous thermal perturbation and mechanical vibration. The behavior of the thermal boundary layer leads to various interesting dynamics such as thermovibrational instabilities, which become particularly ostensive in the absence of gravity. In the present paper, two types of instabilities, Rayleigh-vibrational and parametric instabilities, have been numerically investigated under zero gravity in a two-dimensional configuration using a mathematical model wherein density is calculated directly from the continuity equation. A comparison of experimental observations with numerical simulations is also presented. The peculiarity of the model warrants the investigation of instabilities in a more stringent manner (in terms of higher quench percentage and closer proximity to the critical point), unlike the previous studies wherein the equation of state was linearized around the considered state for the calculation of density, resulting in a less precise analysis. In addition to providing an explanation of the physical causes of these instabilities, we analyze the effect of various parameters on the critical amplitude for the onset of these instabilities. Furthermore, various attributes such as wavelength of the instabilities, their behavior under various factors (quench percentage and acceleration), and the effect of cell size on the critical amplitude are also investigated. Finally, a three-dimensional stability plot is shown describing the type of instability (Rayleigh-vibrational or parametric or both) to be expected for the operating condition in terms of amplitude, frequency, and quench percentage for a given proximity to the critical point.