  This article deals with the variable coefficient thin obstacle problem in
$n+1$ dimensions. We address the regular free boundary regularity, the behavior
of the solution close to the free boundary and the optimal regularity of the
solution in a low regularity set-up. We first discuss the case of zero obstacle
and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the
$C^{1,\alpha}$ regularity of the regular free boundary and derive the leading
order asymptotic expansion of solutions at regular free boundary points. We
further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1}{2}\}}$
regularity of solutions. New ingredients include the use of the Reifenberg
flatness of the regular free boundary, the construction of an (almost) optimal
barrier function and the introduction of an appropriate splitting of the
solution. Important insights depend on the consideration of various intrinsic
geometric structures. Based on variations of the arguments in \cite{KRS14} and
the present article, we then also discuss the case of non-zero and interior
thin obstacles. We obtain the optimal regularity of the solutions and the
regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$
obstacles with $p\in (2(n+1),\infty]$.
