This article examines the stability of a diffusion flame whose leading edge is located near the cold wall to which it attaches. Such leading edges appear in spreading flames, in burner and jet flames, and in flames burning over heterogeneous propellants. The flame model is idealized in order to facilitate analysis and interpretation. The method of analysis follows the examination of a one-dimensional diffusion flame by Vance et al. (2001, Combust. Theory Model., 5, 147). First, a basic state attached-flame problem is defined, which is subjected to small perturbations. The linearized disturbance equations are solved numerically to determine their eigenvalues, which indicate either stability or instability in response to infinitesimal perturbations of the basic state. Many of the one-dimensional model results of Vance et al. (2001) carry over to the two-dimensional case examined here. It is determined that the leading edge dictates the behavior of the remainder of the flame. In addition, the oscillation of the flame leading edge during both flame retreat and flame advance toward the stable position is examined.