Premixed conical flames deform to a quasi-hemispherical shape in the presence of an intense acoustic field. Visualizations show different behaviors of the flame that may be stable or unstable, according to the acoustic power. Effects of the burner geometry are analyzed, and emphasis is put on the influence of the burner exit on the flame shape. Comparisons are made with the acoustic equipotential surfaces at the exit, for two different burners, and mechanisms leading to the modifications of the flame are studied. Using one of the configurations, it was possible to observe strong cells on the flame surface for very high excitation amplitudes. These cells are characteristic of the secondary parametric instability. The instability threshold is given, and the results are compared with studies concerning flames propagating in tubes. In the conditions leading to a stable behavior, where the flame is smooth and quasi-hemispherical, the area of the flame surface is measured. A slight decrease of this area is observed, when the acoustic amplitude is high enough. The displacement speed of the flame front, deduced from measurements of the unburnt gas velocity extrapolated at the flame position, indicates that the flame structure does not seem to be perturbed by the acoustic excitation.