Ignition and extinction characteristics of homogeneous combustion of methane in air near inert surfaces are studied by numerical bifurcation theory for premixed methane/air gases impinging on planar surfaces with detailed chemistry involving 46 reversible reactions and 16 species. One-parameter bifurcation diagrams as functions of surface temperature and two-parameter bifurcation diagrams as functions of equivalence ratio and strain rate are constructed for both isothermal and adiabatic walls. Lean and rich composition limits for ignition and extinction, and energy production are determined from two parameter bifurcation diagrams. For a strain rate of 500 s-1, CH4/air mixtures exhibit hysteresis from approximately 0.5% up to approximately 12.5% and from approximately 5.5% up to approximately 13.5% near isothermal surfaces and adiabatic walls, respectively. Ignition temperature rises with composition from 1,700 to 1,950 K, without a maximum around the stoichiometric ratio. Under some conditions multiple ignitions and extinctions can occur with up to five multiple solutions, and wall quenching, kinetic limitations, and transport can strongly affect flame stability. Flames near the stoichiometric ratio cannot be extinguished by room temperature surfaces for sufficiently low strain rates. The role of intermediates in enhancing or retarding ignition and extinction is studied, and implications of the effect of catalytic surfaces on homogeneous ignition and extinction are discussed. Removal of H atoms and CH3 radicals by wall adsorption can increase extinction and ignition temperature of 6% CH4 in air by up to 300 K for a strain rate of 500 s-1.