Topology transitions of swirl-stabilized flames in a cylindrical combustion chamber are analyzed for lean CH4/H2/air combustible mixtures. It is demonstrated that when the tip of a swirl flame is quenched at the combustor side wall, V to M flame topology transitions are triggered by flashback of the flame front along this wall. An analysis of the power spectral densities of the chemiluminescence luminosity reveals that these topological transitions are not consecutive to the appearance or disappearance of any axisymmetric or helical periodic flow structures interacting with the flame. It is shown that the appropriate space to analyze the stabilization regimes of these swirling flames corresponds to a diagram based on Karlovitz and Lewis numbers. Transitions from V to M flame shapes are found for a critical Karlovitz number below which the flame takes an M shape. The value of this critical Karlovitz number decreases when the Lewis number of the combustible mixture increases. The same type of law is found for M to V shape flame transitions, but this boundary does not coincide with the V to M shape transition boundary. In between these boundaries, the flame takes intermittently V or M shapes randomly. It is then shown that the critical Karlovitz number corresponding to the transition of V to M shape may be expressed by the product of a velocity gradient g calculated with the bulk injection velocity U and a characteristic scale of the outer recirculation zone, the laminar flame thickness α/SL , and the inverse of a vortex induced flame displacement speed St. This displacement speed is related to the maximal azimuthal velocity of the swirling jet at the injector outlet and to the laminar burning velocity. Using this definition of the Karlovitz number, V to M flame topology transitions collapse roughly on the same curve for all flow conditions explored when the swirl number (0.3 ≤ S ≤ 0.68), injection bulk velocity (7 ≤ U ≤ 25 m · s−1), equivalence ratio (0.5 ≤ ϕ ≤ 0.9), and hydrogen enrichment in the fuel blend are modified for rod stabilized and aerodynamically stabilized swirl flames. Results presented in this study might be used to ease predictions of the shape taken by swirl flames interacting with the chamber side wall when the flow operating conditions are modified.