Filtration combustion of lean methane-air mixtures in inert, high-porosity micro-fibrous media was studied experimentally in quartz tubes with different diameters. First, an overall stability diagram was obtained. Between the upstream and downstream propagating regimes, a specific rigid standing wave regime was observed, whereas only one standing wave point existed in other ordinary porous media. A specific instability phenomenon of a combustion wave splitting into two or more parts during wave propagation downstream was also observed in smaller diameter tubes at high filtration velocities. In principle, the results of a conventional two-temperature 1D analytical model captured the main feature of the process obtained in the experiment, but failed to explain the process of flame anchoring to the porous carcass that was observed under variation of the mass flow rate. The possible causes of the failure of the conventional modeling approach were presented.