Self-induced instabilities of laminar premixed flames stabilized over an annular burner have been studied in a set of experiments. A method was developed to determine the stability map of these systems using the response of both the burner and the flame to forced oscillations of the flow. This method is detailed for a well controlled example. The natural unstable motion of the flame is analyzed by measuring velocity fluctuations at the burner outlet, pressure fluctuations inside the burner and variations of the spontaneous light emitted by the flame. The burner's response to external pressure modulations is first characterized without flow and combustion. The flame's response to forced oscillations of the flow at the burner outlet is then used to determine the flame transfer function over the range of frequencies of interest. Using these elements, a mechanism is proposed for the onset of instability, which is shown to result from a coupling of the flame's response to flow oscillations with the bulk resonance mode of the burner. The driving mechanism leading to self-sustained oscillations of the flame front is produced by strong variations of the flame's surface area due to cyclic annihilations of neighboring elements in the flame front. During the collapse of large portions of the flame, a pressure pulse is released, which when properly phased with the bumer acoustics, leads to resonance. A theoretical model for this instability is proposed and it is shown that at resonance, pressure fluctuations inside the burner and heat release fluctuations outside the burner must be in phase, in agreement with the Rayleigh criterion. Modeling predictions are compared to measurements using a diagram combining the acoustical response of the bumer and measurements of the flame transfer functions. Predictions of the potentially unstable flow operating conditions are in good agreement with measurements. A criterion for the onset of instability is derived based on a balance of energy provided to and dissipated by the system. It is found that the gain of the flame transfer function at the resonant frequency of the burner must be sufficient to compensate for the losses of the system. The combined analysis of the burner acoustics and of the flame response to forced modulations of the flow provides a suitable description of instability modes observed experimentally.