We investigate the linear growth and vertical structure of the magnetorotational instability (MRI) in weakly ionized, stratified protoplanetary discs. The magnetic field is initially vertical and dust grains are assumed to be well mixed with the gas over the entire vertical dimension of the disc. For simplicity, all the grains are assumed to have the same radius (a = 0.1, 1 or 3 μm) and constitute a constant fraction (1 per cent) of the total mass of the gas. Solutions are obtained at representative radial locations (R = 5 and 10 au) from the central protostar for a minimum-mass solar nebula model and different choices of the initial magnetic field strength, configuration of the diffusivity tensor and grain sizes. We find that when no grain are present, or they are ≳1 μm in radius, the mid-plane of the disc remains magnetically coupled for field strengths up to a few gauss at both radii. In contrast, when a population of small grains (a = 0.1 μm) is mixed with the gas, the section of the disc within two tidal scaleheights from the mid-plane is magnetically inactive and only magnetic fields weaker than ∼50 mG can effectively couple to the fluid. At 5 au, Ohmic diffusion dominates for z/H ≲ 1 when the field is relatively weak (B ≲ a few milligauss), irrespective of the properties of the grain population. Conversely, at 10 au this diffusion term is unimportant in all the scenarios studied here. High above the mid-plane (z/H ≳ 5), ambipolar diffusion is severe and prevents the field from coupling to the gas for all B. Hall diffusion is dominant for a wide range of field strengths at both radii when dust grains are present. The growth rate, wavenumber and range of magnetic field strengths for which MRI-unstable modes exist are all drastically diminished when dust grains are present, particularly when they are small (a ∼ 0.1 μm). In fact, MRI perturbations grow at 5 au (10 au) for B ≲ 160 mG (130 mG) when 3 μm grains are mixed with the gas. This upper limit on the field strength is reduced to only ∼16 mG (10 mG) when the grain size is reduced to 0.1 μm. In contrast, when the grains are assumed to have settled, MRI-unstable modes are found for B ≲ 800 mG at 5 au and 250 mG at 10 au. Similarly, as the typical size of the dust grains diminishes, the vertical extent of the dead zone increases, as expected. For 0.1 μm grains, the disc is magnetically inactive within two scaleheights of the mid-plane at both radii, but perturbations grow over the entire section of the disc for grain sizes of 1 μm or larger. When dust grains are mixed with the gas, perturbations that incorporate Hall diffusion grow faster, and are active over a more extended cross-section of the disc, than those obtained under the ambipolar diffusion approximation. Note that the stabilizing effect of small dust grains (e.g. a = 0.1 μm) is not strong enough to completely suppress the perturbations. We find, in fact, that even in this scenario, the magnetic field is able to couple to the gas and shear over a range of fluid conditions. Despite the low-magnetic coupling, MRI modes grow for a range of magnetic field strengths and Hall diffusion largely determines the properties of the perturbations in the inner regions of the disc.