In the singlet ground-state systems CsFeCl3 and CsFeBr3 a large single-ion anisotropy causes a singlet ground state and a doubly degenerate doublet as the first excited states of the Fe2+ ion. In addition the magneteic interaction is anisotropic being much larger along the z axis than perpendicular to it. Therefore, these quasi-one-dimensional magnetic model systems are ideal to demonstrate unique correlation effects. Within the framework of the correlation theory we derive the expressions for the excitation spectrum. When a magnetic field is applied parallel to the z axis both substances have phase transitions to commensurate long-range order. In CsFeCl3 this transition is preceded by two transitions to incommensurate structures. The calculated fluctuation effects can indeed explain the experimentally detected incommensurate order in CsFeCl3, and also the absence of that in CsFeBr3. A sophisticated numerical and graphical method leads to a self-consistent determination of the induced magnetization and the quadrupole moment as well as to the determination of the excitation spectrum for CsFeBr3 and CsFeCl3 as a function of the magnetic field. For magnetic fields below the phase transition the experimental data can be excellently described by the self-consistent random-phase approximation results. For magnetic fields near the critical magnetic field only qualitative conclusions can be obtained. Numerical results for the critical scattering, the correlation lengths, and the specific heat, which are based on the analysis of the first-moment frequency, support the supposition that the field-induced transition is approaching a second-order phase transition.