The replacement of hydrocarbon fuels in energy production and aeronautical applications by hydrogen raises issues linked to its enhanced reactivity, broad flammability limits and augmented burning velocity. Hydrogen flames may also feature specific sensitivity to acoustic disturbances that may lead to combustion instabilities. This problem is examined in the present work in the case of a single flame formed by a swirling injection unit where hydrogen is injected in crossflow to enhance mixing to reduce hotspots. Self-sustained oscillations (SSOs) are observed using optical diagnostics and acoustic pressure measurements. It is shown that changes in flame shape during one oscillation cycle feature a periodic attachment and detachment of the flame from the injector, and aerodynamic disturbances enhancing flame wrinkling. The influence of thermal power and global equivalence ratio is investigated and it is shown that unstable oscillations arise when the system is operated in an intermediate thermal power regime where the flame alternates between lifted and attached configurations. Investigations of the effect of swirl number and recess of hydrogen injection with respect to the chamber backplane show that the unstable range is diminished when the recess takes large values and also when the swirl number is increased.