We examine anchored edge-flames in non-premixed combustion for which the Lewis number of the fuel is large. Steady two-dimensional solutions are constructed, both stable and unstable, by using a modified time-integration strategy. These calculations define a static detachment Damkohler number. Real-time integrations reveal oscillating instabilities if the Damkohler number D is small enough, and these instabilities are also examined by linear perturbation of the steady solutions. The eigenfunctions constructed in this way make clear that the instability is confined to the edge, and does not affect the ID flame trailing from the edge. Dynamic detach-ment of an oscillating flame occurs in one of two ways, depending on the boundary conditions used in the model: as D is decreased, the period and amplitude of the oscillation grow and approach infinity as D approaches a critical value greater than the static detachment value; or, the period and amplitude remain bounded, and dynamic detachment occurs at the same value of D as for static detachment.