  We study the Muskat problem on the half-plane, which models motion of an
interface between two fluids of distinct densities (e.g., oil and water) in a
porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g.,
bedrock). We show that unlike on the whole plane, finite time stable regime
singularities do arise in this setting, including from arbitrarily small smooth
initial data. To obtain this result, we establish maximum principles for both
the $L^2$ norm and the slope of solutions in this model, as well as develop a
general local well-posedness theory in the companion paper [21].
