  We obtain local well-posedness for 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). Our result allows for the interface to touch
the bottom, and hence applies to the important scenario of the heavier fluid
invading a region occupied by the lighter fluid along the impermeable layer. We
use this result in the companion paper [24] to prove existence of finite time
stable regime singularities in this model, including for arbitrarily small
initial data. We do not require the interface and its derivatives to vanish at
$\pm\infty$ or be periodic, and even allow it to be $O(|x|^{1-})$, which is an
optimal bound on the power of growth. We also extend our results to the cases
of the Muskat problem on the whole plane and on horizontal strips, where all
previous works did impose such limiting requirements.
