I will first present a convergence result for solutions of Allen-Cahn type systems with a multiple-well potential involving the usual fractional Laplacian in the regime of the so-called nonlocal minimal surfaces. In the singular limit, solutions converge in a certain sense to stationary points of a nonlocal (or fractional) energy for partitions of the domain with (in general) non homogeneous surface tensions. Then I will present partially regularity results and open questions concerning the limiting problem underlying the new features compared to classical minimal partition problems. This talk is based on joint works with Thomas Gabard. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and the PRIN project 2022PJ9EFL, Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations.