In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term \( \sigma \). When \( \sigma \) is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if \( \sigma \) is continuous, then minimizers are of class \( C^1 \) along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.