Extensions of the minimal supersymmetric standard model (MSSM) with additional singlet scalar fields solve the important µ-parameter fine-tuning problem of the MSSM.We compute and compare the neutral Higgs boson mass spectra, including one-loop corrections, of the following MSSM extensions: next-tominimal supersymmetric standard model (NMSSM), the nearly-minimal supersymmetric standard model (nMSSM), and the U(1)'-extended minimal supersymmetric standard model (UMSSM) by performing scans over model parameters. We find that the secluded U(1)'-extended minimal supersymmetric standard model (sMSSM) is identical to the nMSSM if three of the additional scalars decouple. The dominant part of the one-loop corrections are model independent since the singlet field does not couple to MSSM particles other than the Higgs doublets. Thus, model-dependent parameters enter the masses only at tree level. We apply constraints from LEP bounds on the standard model and MSSM Higgs boson masses and the MSSM chargino mass, the invisible Z decay width, and the Z - Z' mixing angle. Some extended models permit a Higgs boson with mass substantially below the SM LEP limit or above theoretical limits in the MSSM.Ways to differentiate the models via masses, couplings, decays and production of the Higgs bosons are discussed.