Three-dimensional near-equilibrium potential energy surfaces and dipole moment functions have been calculated for the ground states of NCO- and CNO-, using the coupled cluster method with single and double substitutions augmented by a perturbative estimate of triple excitations [CCSD(T)] with a quadruple zeta basis set consisting of 150 contracted Gaussian type orbitals. The corresponding equilibrium bond distances at their linear geometries are r(e)(NC) = 1.1934 Angstrom and r(e)(CO) = 1.2306 Angstrom for NCO-, and r(e)(CN) = 1.1866 Angstrom and r(e)(NO) = 1.2741 Angstrom for CNO-. Full three-dimensional variational calculations have also been carried out using the CCSD(T) potential energy and dipole moment functions to determine the rotation-vibrational energy levels and dipole moment matrix elements for both NCO- and CNO-. The predicted band origin of the nu(3) band in the NCO- isomer (2114.4 cm(-1)) agrees well with the gas phase diode laser infrared result (2124.4 cm(-1)). The variational analysis suggests possible revisions in the assignment of the two experimentally observed hot bands which are affected by Fermi resonance. The calculated dipole moments of NCO- and CNO- in their ground vibrational states are 1.504 and 1.482 D, respectively. The CCSD(T) method with a triple zeta basis set was employed to more broadly explore the isomerization path between the two isomers. In agreement with previous lower level calculations a broad and shallow minimum corresponding to a cyclic oxazirine form was found. The immediate vicinity of this local minimum in the potential energy surface and those of the two saddle points separating it from the linear isomers were further refined using the same quadruple zeta basis used for the two near-equilibrium regions. The equilibrium structures and energies of the two neutral isomers, NCO, and CNO, were also calculated at the same level of theory. For NCO, whose photoelectron spectrum has recently been studied in detail, the predicted electron affinity and neutral-ion bond distance changes agree well with the photoelectron results. The reported spectroscopic structure of NCO, however, is not supported by the present CCSD(T) calculations.