We study the Saffman-Taylor instability in a non-Brownian suspension by injection of air. We find that flow structuration in the Hele-Shaw cell can be described by an effective viscosity depending on the volume fraction. When this viscosity is used to define the control parameter of the instability, the classical finger selection for Newtonian fluids is recovered. However, this picture breaks down when the cell thickness is decreased below approximatively 10 grain sizes. The discrete nature of the grains plays also a determinant role in the the early destabilization of the fingers observed. The grains produce a perturbation at the interface proportional to the grain size and can thus be considered as a "controlled noise". The finite amplitude instability mechanism proposed earlier by Bensimon et al. allows to link this perturbation to the actual values of the destabilization threshold.