Abstract.  Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are "(Arnold) resonance horns" emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions -- different subregions corresponding to maps having different numbers of periodic orbits. The existence of subregions having more than the "usual" one pair of attracting and repelling periodic orbits implies the existence of "extra folds" in the corresponding surface of periodic points in the cartesian product of the phase and parameter spaces. The existence of more than one attracting and one repelling periodic orbit is shown to be generic. For some of the families we create, the resulting parameter plane bifurcation pictures appear in shapes we call "Arnold flames." Results apply both to circle maps and forced oscillator maps.