  Coplanar model states for applications of the coupled cluster method (CCM) to
problems in quantum magnetism are those in which all spins lie in a plane,
whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones
in which all the spins do not lie in any single plane. Here we extend the CCM
to non-coplanar / 3D model states and we present results for three cases: (a)
the spin-half one-dimensional Ising ferromagnet in an applied transverse
magnetic field (as an exactly solvable test model to use as a yardstick for the
viability and accuracy of our new methodology); (b) the spin-half
triangular-lattice Heisenberg antiferromagnet in the presence of an external
magnetic field; and (c) the spin-$S$ triangular-lattice {\it XXZ}
antiferromagnet in the presence of an external magnetic field, for the cases
$\frac{1}{2} \leq S \leq5 $. For 3D model states the sets of algebraic CCM
equations for the ket- and bra-state correlation coefficients become
complex-valued, but ground-state expectation values of all physical observables
are manifestly real numbers, as required. Excellent correspondence is seen with
the results of other methods, where they exist, for these systems. CCM results
demonstrate explicitly that coplanar ordering is favoured over non-coplanar
ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all
values of the applied external magnetic field, whereas for the anisotropic {\it
XXZ} model non-coplanar ordering can be favoured in some regions of the
parameter space. Specifically, we present a precise determination of the
boundary (i.e., the critical value of the {\it XXZ} anisotropy parameter
$\Delta$) between a 3D ground state and a coplanar ground state for the {\it
XXZ} model for values for the external magnetic field near to saturation, for
values of the spin quantum number $S \leq 5$.
