In general, the eigenmodes of a cold inhomogeneous magnetoplasma have magnetic field perturbations with components both parallel and transverse to the background magnetic field direction. Recently, constraints have been derived which the medium must satisfy for the field perturbation to be either entirely compressional, or entirely transverse. Here these criteria are investigated to yield all planar and axisymmetric magnetic field configurations allowing pure decoupled transverse and compressional oscillations. For example, in an axisymmetric dipole field the azimuthal field perturbations decouple exactly, but solely poloidal transverse ones do not. The magnetic field geometry and background plasma density for the most useful decoupled solutions are listed in a table at the end of the paper. ¿ American Geophysical Union 1991