  We consider the one-phase Muskat problem modeling the dynamics of the free
boundary of a single fluid in porous media. We prove local well-posedness for
fluid interfaces that are general curves and can have singularities. In
particular, the free boundary can have acute angle corners or cusps. Moreover,
we show that isolated corners/cusps on the interface must be rigid, meaning the
angle of the corner is preserved for a finite time, there is no rotation at the
tip, the particle at the tip remains at the tip and the velocity of that
particle at the tip points vertically downward.
