The most common designs of magnetic bearing utilize magnetic normal stress to exert net lateral force between the bearing rotor and the bearing stator. For such bearings, the specific load capacity (defined as the ratio of maximum bearing force to total bearing self-weight) is intrinsically limited by the magnetic saturation of iron at flux densities in the order of 2 Tesla. Specific load capacities of 100: 1 are about at the limit of what is achievable with conventional designs. A new class of magnetic bearings has been proposed with a view to achieving substantially higher specific load capacity values. These bearings achieve net lateral force between bearing rotor and stator as a result of magnetic shear stress acting over a number of parallel annular airgaps. The airgaps are defined as the spaces between discs fixed to the stator and discs fixed to the rotor. Mechanical stability is one of the key limitations affecting the design of such bearings. Two separate effects conspire to attempt to destabilize the discs: out-of-plane magnetic negative stiffness and in-plane loading. The elastic stiffness of the discs must be sufficiently high to prevent instability. This paper presents methods for establishing where the stability boundaries lie.