Free boundary problems naturally arise in a variety of models from material science, particularly in connection with phase transitions and interfacial phenomena. In this talk, we discuss a class of variational problems that lead to singular PDEs in heterogeneous and irregular media. We focus on models whose intrinsic singularities are shaped by the medium itself, so that variations in the underlying structure directly influence the geometric behavior of solutions in a highly nonlinear fashion. On the analytical side, this leads to the development of a layered regularity theory, designed to remain robust across different regimes. In more structured settings, we establish $C^{1,\alpha}$-regularity of the free boundary, showing how increased organization of the medium is reflected in the qualitative behavior of solutions and their interfaces.