  We consider steady surface waves in an infinitely deep two--dimensional ideal
fluid with potential flow, focusing on high-amplitude waves near the steepest
wave with a 120 degree corner at the crest. The stability of these solutions
with respect to coperiodic and subharmonic perturbations is studied, using new
matrix-free numerical methods. We provide evidence for a plethora of
conjectures on the nature of the instabilities as the steepest wave is
approached, especially with regards to the self-similar recurrence of the
stability spectrum near the origin of the spectral plane.
