  We present a linear analysis of the vertical structure and growth of the
magnetorotational instability in stratified, weakly ionised accretion discs,
such as protostellar and quiescent dwarf novae systems. The method includes the
effects of the magnetic coupling, the conductivity regime of the fluid and the
strength of the magnetic field, which is initially vertical. The conductivity
is treated as a tensor and assumed constant with height. We obtained solutions
for the structure and growth rate of global unstable modes for different
conductivity regimes, strengths of the initial magnetic field and coupling
between ionised and neutral components of the fluid. The envelopes of
short-wavelenght perturbations are determined by the action of competing local
growth rates at different heights, driven by the vertical stratification of the
disc. Ambipolar diffusion perturbations peak consistently higher above the
midplane than modes including Hall conductivity. For weak coupling,
perturbations including the Hall effect grow faster and act over a more
extended cross-section of the disc than those obtained using the ambipolar
diffusion approximation. Finally, we derived an approximate criterion for when
Hall diffusion determines the growth of the magnetorotational instability. This
is satisfied over a wide range of radii in protostellar discs, reducing the
extent of the magnetic `dead zone'. Even if the magnetic coupling is weak,
significant accretion may occur close to the midplane, rather than in the
surface regions of weakly-ionised discs.
