A new methodology is developed to study the stability of multiple solutions and the onset of oscillations of distributed flames modeled with detailed chemistry and multicomponent transport. This methodology is applied to premixed hydrogen/air mixtures impinging onto an inert isothermal surface. In particular, the local stability of the extinguished, ignited, partially ignited, and intermediate branches is determined on-the-fly as stationary solutions are computed. Hopf bifurcation points appear only in the fuel-lean and fuel-rich regime, near the edges of a nonextinction regime. Harmonic, relaxation, and complex mode self-sustained oscillations can occur depending on surface temperature, and multistage ignitions are found, varying from three-stage to six-stage ignitions. In the presence of a Hopf bifurcation, it is found that ignition can be oscillatory, and extinction can be oscillatory at an infinite period saddle-loop bifurcation or coincident with a Hopf bifurcation. The implications of such behavior for extinction theory are briefly discussed. It is shown that Hopf bifurcation has a kinetic origin but is affected by the heat of reactions as the composition approaches a thermally nonextinction regime. For strong flames, thermal feedback destroys oscillatory dynamics. Sensitivity analysis of Hopf bifurcation shows that the termination reaction H + O-2 + M --> HO2 + M plays an important role in the birth of oscillatory dynamics and that diffusion of H2O is also significant.