In this paper, the effect of geometry on the dynamics of the hydrogen–air flame stabilised on the surface of a burner is investigated numerically. Two main regimes of combustion front stabilisation near the cylindrical burner: attached and detached flames are investigated and compared to the planar case. The neutral stability boundary for diffusive-thermal instability is studied in detail. The attached flame regime is characterised by the fact that for rich mixture compositions the flame becomes less stable in comparison to the combustion front stabilised near a planar burner. The detached flame regime is influenced by the divergent outflow, which significantly perturbs the flow field in the product regions of the flame. In contrast to the detached regime in the attached regime curvature stabilises the flame front. The flame properties investigated in this study are extremely sensitive to chemical kinetic models. The results and observations made can be used to support experimental design to set-up experiments for additional validation of detailed mechanisms of chemical reactions.