  We provide a rather complete description of the sharp regularity theory for a
family of heterogeneous, two-phase variational free boundary problems,
$\mathcal{J}_\gamma \to $ min, ruled by nonlinear, $p$-degenerate elliptic
operators. Included in such family are heterogeneous cavitation problems of
Prandtl-Batchelor type; singular degenerate elliptic equations; and obstacle
type systems. The Euler-Lagrange equation associated to $\mathcal{J}_\gamma$
becomes singular along the free interface $\{u= 0\}$. The degree of singularity
is, in turn, dimed by the parameter $\gamma \in [0,1]$. For $0< \gamma < 1$ we
show local minima is locally of class $C^{1,\alpha}$ for a sharp $\alpha$ that
depends on dimension, $p$ and $\gamma$. For $\gamma = 0$ we obtain a
quantitative, asymptotically optimal result, which assures that local minima
are Log-Lipschitz continuous. The results proven in this article are new even
in the classical context of linear, nondegenerate equations.
