Edge flames have been interested as a basic structure that is concerned to flame stabilization and re-ignition of non-premixed flames. The edge flame consists of a lean premixed flame, a rich premixed flame, and a diffusion flame. In order to investigate fundamental structures of the edge flames at the conditions near the flammability limits, edge flames were stabilized within a heated narrow channel. Highly diluted partially premixed methane was used, and the flow rates of air and the partially premixed mixture were controlled. Various flame behaviors, including a transition between ordinary edge flames and premixed flames, were observed. Flame stabilization characteristics were examined as well. All flame stabilization conditions in this study showed a similar trend: characteristic time scales were inversely proportional to the equivalence ratio defined at the burner inlet. Finally, an interesting flame structure having a weak diffusion branch enveloped by a closed premixed branch was found near the flammability limits even in a fuel-air mixing layer. This structure was named as a "flame-drop" and the importance of this structure was first suggested.