  We prove local higher-order asymptotics for extreme water waves with
vorticity near stagnation points. We obtain that the behaviour of solutions and
their regularity depend substantially on the vorticity. In particular, we show
that extreme waves with a negative vorticity distribution have concave profiles
near the crest. Our approach is based on new regularity results and asymptotic
analysis of the corresponding nonlinear problem in a half-strip. Our main
result is local and therefore is valid for a broad range of problems, such as
for waves with a piecewise constant vorticity, stratified waves, flows with
counter-currents or waves on infinite depth.
