The Poiseuille-Rayleigh-Bénard problem, involving the onset of thermoconvective structures in channels heated from below, was the subject of many theoretical, numerical and experimental studies for incompressible flows or perfect gas. However, to the authors' knowledge, this problem was never studied for supercritical fluids. The objective of this paper is to study the influence of the specific properties of such fluids on thermoconvective instability phenomena compared with those observed in the perfect gas case. The effect of the distance to the critical point is also investigated. The numerical approach used is based on the Navier-Stokes equations in the framework of the low Mach number approximation.