We discuss edge-flames in the context of the plane counterflow of air and nitrogen-diluted hydrogen. For sufficiently dilute hydrogen streams, strong cellular instabilities arise which affect the dynamics of the edge-flames and, in the long term, give rise to residual cellular structures. Depending on the Damkohler number, solutions are constructed corresponding to: an advancing edge trailing a smooth flame; an advancing edge trailing a stationary periodic cellular structure, a warp of flame-strings; a finite array of flame-strings which drift apart from each other at a decreasing rate; a single stationary flame-string; dual stationary flame-strings; triple stationary flame-strings. Many of these solutions are sublimit in the sense that they exist for Damkohler numbers smaller than the one-dimensional quenching value defined for the counterflow configuration. The connection of these solutions to the linear stability of the one-dimensional system is discussed, and evidence of a subcritical bifurcation is presented. Comparisons are made with experimental results obtained by Ronney.