We present numerical results on the flame attachment in the downstream vicinity of the co-flow injector lip that separates the reactive fluids at injection. Two stability diagrams show the domains where the flame is anchored, blown off, or extinguished, in terms of separating plate thickness and injection velocities of both fluids. Different anchoring modes—stagnation point counter-flow holding or edge flame anchorage—are described, depending particularly on the plate rim thickness.