Unsteady non-premixed flame structures of methane/oxygen mixtures are investigated at supercritical pressures, representative of liquid rocket engines combustion chamber operating conditions. A general-fluid formulation of the flamelet equations is used and deviations from ideality of the thermodynamic properties are taken into account by means of a computationally efficient cubic equation of state written in a general three-parameter fashion. The effects of pressure and scalar dissipation rate are investigated in the context of prototypical unsteady laminar flame configurations, such as autoignition and re-ignition/quenching. In auto-igniting flamelets, real gas effects are observed to influence different flame regions depending on the thermodynamic pressure. Moreover, the mixture ensuing from methane oxidation is never observed to reach a saturated (two-phase) thermodynamic region. Re-ignition and quenching phenomena are analyzed using a time dependent forcing function for the scalar dissipation rate, in order to investigate the response of the real gas flame structures to typical turbulent perturbations. The role of pressure on the critical strain values that a real gas laminar flame can sustain without quenching is investigated and compared to its ideal gas counterpart.