The limits of existence of a steady planar methane-air flame stabilized on a flat porous burner at normal and elevated pressure (2, 4 and 6 bar) have been experimentally and numerically investigated. In particular, the critical conditions for the blow-off and diffusive-thermal oscillations have been determined in the plane of parameters: mass flow rate vs. equivalence ratio. The Hopf frequency of oscillations is measured at the diffusive-thermal oscillation boundary. The results of numerical simulations, undertaken with the use of detailed reaction mechanisms, such as GRI, FFCM, USC II, SanDiego, and Aramco, show that, despite the good qualitative agreement with the experimental data, the relative quantitative difference between the numerical simulations and the experimental measurements is quite large. It is of the order of several tens of percent and is especially evident when the measurements are performed away from stoichiometry and under high pressures. In order to verify and validate detailed reaction mechanisms, in addition to the standard tests such as measurement of laminar burning velocity, ignition delay time and extinction strain rate, it is necessary to obtain a wider range of experimental data. It is especially important at elevated pressures and high temperatures. Determining the characteristics of the diffusion-thermal oscillations is a suitable way to achieve this. Novelty and Significance Statement For the first time, we experimentally found the critical conditions for the blow-off and onset of diffusive thermal pulsating instabilities, as well as the characteristics of diffusive thermal oscillations for the burner stabilized methane-air flames at elevated pressure. These data were compared for the first time with the predictions of several detailed reaction mechanisms to verify their performance under such conditions. The novel findings reported in this work on the regions of existence of stable combustion regimes are significant for the design of practical burners, while the data on the conditions and characteristics of the critical phenomena will facilitate the development of accurate and efficient mechanism of methane combustion.