We consider the two-dimensional water wave equation which is a model of ocean waves. The water wave equation is a free boundary problem for the Euler equation where we assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the case of zero surface tension, we show that the singular solutions recently constructed by Wu are rigid. In the case of non-zero surface tension, we construct an energy functional and prove an a priori estimate without assuming the Taylor sign condition. This energy reduces to the energy obtained by Kinsey and Wu in the zero surface tension case for angled crested water waves. We show that in an appropriate regime, the zero surface tension limit of our solutions is the one for the gravity water wave equation which includes waves with angled crests.