An experimental study is carried out to analyze the effects of swirl on the structure and stabilization of methane non-premixed oxygen-enriched flames above a coaxial injector in which the two streams are eventually swirled. The mean position of the flame and the liftoff height above the injector lips are investigated with OH* chemiluminescence images. The oxygen enrichment, the momentum flux ratio between the two coflows, the swirl level inside the central jet, and the swirl level in the annular jet are varied over a large range of operating conditions. It is found that, in the absence of swirl in the central stream, the flame is always attached to the lips of the internal injection tube. As the inner swirl level increases, the flame front located at the lips of the internal injection tube disappears. When the annular swirl level is high enough to create a central recirculating bubble, the flame detaches from the nozzle rim and remains lifted at a finite distance above the injector. Increasing the oxygen concentration shifts this transition to smaller momentum flux ratios and smaller annular swirl levels. The liftoff distance can be finely tuned and depends on the inner and outer swirl levels, and on the momentum flux ratio between the two coaxial streams. It is shown that this feature depends neither on the confinement of the injector nor on the thermal stress exerted by the hot burnt gases on the injector back plane. About 1000 configurations were investigated that could be classified into only four distinct stabilization modes, in which the flame structure was shown to follow a similar pathway when the momentum flux ratio between the two streams, the swirl level in the central and external streams, and the quarl angle of the annular stream are varied. It is finally shown how these limits are altered when the oxygen concentration in the annular oxidizer stream is varied from air to oxygen-enriched operation.