Three-dimensional simulations of a non-premixed microburner (channel height L-z = 0.75 mm) are used to study flame structure and stability, for both H-2-O-2 and CH4-O-2 mixtures, matching conditions of previously reported experiments. Thermal quenching and slow diffusive mixing lead to incomplete combustion and, for some flow rates, steady flame streets form in the channel for CH4-O-2, matching experimental observations. Still smaller-scale burners, with channel heights L-z = 0.375 mm and L-z = 0.25 mm, are also simulated, and flame streets are seen even for H-2-O-2 cases due to the strong thermal quenching. A wall-chemistry model is used to assess the importance of wall quenching of H and O radicals. Wall recombination kinetics weaken the flames, reduce temperature by over 100 K, significantly reduce the length of the flame diffusion tails, and reduce overall combustion completeness. The basic mechanisms observed to be important in the microburner channel are included in an analogous one-dimensional diffusion-flame model, which includes a heat-loss factor motivated by the full burner. For similar conditions to the microburner and high heat loss, the solution oscillates sufficiently strongly to extinguish the flame, as observed in some microburner cases. For more modest thermal quenching, the oscillations persist and are analogous to the stable flame streets seen in the microburner. (C) 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.