Instabilities of premixed and diffusion hydrogen/air flames near surfaces are modeled using complex gas-phase reaction chemistry and numerical bifurcation theory. Steady-state multiplicities are computed using a combined dynamically adaptive multiple weights are-length continuation algorithm and Newton's method, and their dynamics is examined through local stability analysis and direct numerical integration. For premixed flames, it is found that Hopf bifurcations exist over a wide range of pressure, and the pressure-temperature stability diagram exhibits many similarities to experimental results and simulations in a continuous-stirred tank reactor. It is shown that reaction exothermicity is not a prerequisite for oscillations but can expand the regime over which oscillations are observed. We report oscillations for diffusion flames at atmospheric pressure, for the first time, as the surface fuel flow rate decreases toward the thermally quenched flammability limit. Overall, the bifurcation behavior of a diffusion flame exhibits a remarkable similarity to that of a premixed flame by an appropriate choice of parameters. Implications for oscillations in condensed fuels prior to flame extinction and future research directions are briefly discussed.