  This is a study of singular solutions of the problem of traveling gravity
water waves on flows with vorticity. We show that, for a certain class of
vorticity functions, a sequence of regular waves converges to an extreme wave
with stagnation points at its crests. We also show that, for any vorticity
function, the profile of an extreme wave must have either a corner of
$120^\circ$ or a horizontal tangent at any stagnation point about which it is
supposed symmetric. Moreover, the profile necessarily has a corner of
$120^\circ$ if the vorticity is nonnegative near the free surface.
