We connect classical partial regularity theory for elliptic systems to Nonlinear Potential Theory of possibly degenerate equations. More precisely, we find a potential theoretic version of the classical ε -regularity criteria leading to regularity of solutions of elliptic systems. For non-homogenous systems of the type - div a ( D u ) = f , the new ε -regularity criteria involve both the classical excess functional of D u and optimal Riesz type and Wolff potentials of the right hand side f . When applied to the homogenous case - div a ( D u ) = 0 such criteria recover the classical ones in partial regularity. As a corollary, we find that the classical and sharp regularity results for solutions to scalar equations in terms of function spaces for f extend verbatim to general systems in the framework of partial regularity, i.e. optimal regularity of solutions outside a negligible, closed singular set. Finally, the new ε -regularity criteria still allow to provide estimates on the Hausdorff dimension of the singular sets.