The partial differential equation (PDE) describing the dynamics of hydrodynamically unstable planar flame front has exact pole solutions which satisfy a set of ordinary differential equations (ODEs). This set of ODEs prohibits the creation of new poles in the complex plane, or the appearance of cusps in the physical space, as observed experimentally. The contribution of this paper is to show that most exact pole solutions are unstable solutions for the PDE. Even the one-peak, coalescent solutions (whose number of poles is maximal) is unstable as soon as the number of poles exceeds a certain (rather small) threshold. As the size of the domain increases, the front undergoes bifurcations which can be described as follows: the one-pole, one-peak coalescent solution is neutrally stable for small intervals. As the length of the interval increases, it becomes unstable and the two-pole one-peak coalescent solution is neutrally stable. For larger intervals, the two-pole solution is unstable, the three-pole solution becomes stable. As the interval length increases further, the steady one-peak, coalescent solutions are no longer stable and bifurcations to unsteady states occur.