We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle L. Especially we study minimal singular metrics of L, metrics of L with the mildest singularities among singular Hermitian metrics of L whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of L. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semipositive curvature.