A unified view on the interfacial instability in a model of aluminium reduction cells in the presence of a uniform, vertical, background magnetic field is presented. The classification of instability modes is based on the asymptotic theory for high values of parameter β, which characterises the ratio of the Lorentz force based on the disturbance current, and gravity. It is shown that the spectrum of the travelling waves consists of two parts independent of the horizontal cross-section of the cell: highly unstable wall modes and stable or weakly unstable centre, or Seles modes. The wall modes with the disturbance of the interface being localised at the sidewalls of the cell dominate the dynamics of instability. Seles modes are characterised by a distributed disturbance over the whole horizontal extent of the cell. As β increases these modes are stabilized by the field.