Supercritical natural circulation loops (NCLs) promise passive cooling for critical systems like nuclear reactors and solar collectors, eliminating the need for mechanical pumps. However, instabilities similar to those seen in two-phase systems can emerge in supercritical NCLs, leading to undesirable oscillatory behaviour, marked by system-wide fluctuations in density, temperature, pressure, and flow rate. This study investigates the stability of NCLs at supercritical pressures (73.7≤p≤110.0bar) using CO₂ in an experimental setup with vertical cooling and vertically adjustable heaters to control convective flow rates and to oppose flow reversal. Oscillations were found to originate in the heater of the NCL, and demonstrated a high sensitivity to the thermodynamic state and proximity to the pseudo-critical line of the system. Increased mass flow rates and added resistance upstream of the heater suppressed the oscillations, while increased pressures and reduced heating rates dampened them. A static model which takes into account the non-ideality of the heat exchangers is introduced to assess the presence of multiple steady states. The system is concluded to be statically stable, and the oscillations are considered to be dynamically induced. In particular, the modulation of the NCL velocity by the traversal of the current oscillations in density is assumed to periodically re-incite non-ideality in the heater. These findings intend to refine our understanding of the stability boundaries in NCLs, to ensure a safer operation of prospective passive cooling and circulation systems employing fluids at supercritical pressure.