Rogue waves turn out to be not that rare

[jehardiman]

For a non-breaking, regular, wave train the angle between the face and back cannot be less than 120 degrees. At this angle, the peak forms a cusp that cannot continue to be supported. See any of the references I have already cited.

For irregular waves, the matter is much more complicated, as the slope is very dynamic. It is possible, due to group speed dynamics to have...for a very short period... a vertical wave face or back. However a face like this cannot be supported by the fluid and the group speeds soon cause seperation. This dynamic causes the rolling and plunging crests often seen in photos of steep seas.

Edit to add, the above was for deep water waves. For surf (i.e. waves entering shoaling water), that is a whole different topic...