Hydrogen combustion is currently being considered as a possible means of decarbonization of power generation and air transportation. However, hydrogen raises difficult issues including that of NOx emissions linked to the high flame temperature and that of flashback caused by its high burning velocity. Innovative injection schemes are required allowing operation under lean conditions by enhancing mixing to avoid hotspots. In addition, the flame needs to be stabilized at a distance from solid boundaries to reduce thermal loads and should offer a low sensitivity to disturbances and instability. These issues are investigated with systematic experiments carried out in a single injector laboratory-scale combustor operating at atmospheric pressure, a configuration that was already used to examine the dynamics of spray flames. The atomizer is replaced by a special unit (CFI-X) that allows gaseous hydrogen injection in crossflow and mixing with the air stream set in rotation by a radial swirler. The flame behavior formed by this device is examined as a function of thermal power and global equivalence ratio. The injection head recess is also varied to change the level of mixing in the system and see how it influences thermo-acoustic instabilities. Flame structures visualized through OH* light emission images are used to classify the various flame patterns and distinguish domains where the flame is detached from the injector outlet. Variations of the injection head recess with respect to the outlet, swirl number and chamber length provide a comprehensive view of the operating regimes. It is found that the flames remain attached to the injector at low power and that detached flames are obtained in the high-power range. Dynamical phenomena leading to oscillations coupled by longitudinal chamber modes are observed in an intermediate thermal power range. The corresponding data are interpreted with a stability analysis that defines instability bands for the product of a time delay by the frequency of oscillation. It is found that unstable flames belong to the first of these bands. It is also observed that detached flames that are the most interesting in terms of thermal loads and mixing may also belong to the band of instability but do not give rise to oscillations and are therefore less sensitive to instability.