A solution is presented for the Koebe problem of characterizing compacta that generate minimal domains. This, in turn, makes it possible to describe the zero-sets for the class of regular functions with bounded Dirichlet integrals, and for its generalization in the Rodin-Sario-Hedberg sense as removable sets in the corresponding modulus problem. In §1 we use the limiting curve method to prove assertions about “continuity” of the \(p\)-modulus of a family of compound curves and to establish that this \(p\)-modulus is equal to the \(p\)-capacity of the corresponding condenser, which strengthens known results of J. Hesse [Ark. Mat. 13, 131-144 (1975; Zbl 0302.31009) and H. Yamamoto, Hiroshima Math. J. 8, 123-150 (1978; Zbl 0389.31003)] in this direction. An application of the results obtained to the Dirichlet problem is given. In §2 we investigate the projection \({\mathcal H}^1\)-measure of a curve, study the properties of compacta generating \(p\)-normal domains, and, in particular, present a solution of the Koebe problem mentioned above. In §3 we consider local characteristics of zero-sets for \(FD^p\) that are close in form to results of B. Fuglede on \(p\)-exceptional sets [Acta Math. 98, 171-219 (1957; Zbl 0079.27703)]. The class of \(q\)-removable sets for the family of irreducible continua in \(\mathbb{R}^n\) is shown to be equal to the class of \(NH_p\)-compacta.