A study of instabilities in planar flows produced by the presence of a parallel penetrable porous obstruction is presented. The case considered is a flow between parallel plates partially obstructed by a porous medium. The most unstable perturbation modes are obtained solving numerically the eigenvalue problem derived from the linear stability analysis of the two-dimensional Navier–Stokes equations applied to the geometry of interest. The analysis leads to an extended Orr–Sommerfeld equation including a porous term. It was found that the ratios of the permeability and depth of the obstruction with respect to the free flow layer depth are the relevant parameters influencing the stability margin and the structure of the most unstable modes. To validate the conclusions of the theoretical analysis, an experiment with air flowing through a channel semi-obstructed by a regular array of cylindrical wires was performed. The critical Reynolds number, which was determined by measuring the amplitude of velocity fluctuations at the interface of the porous medium, agrees with the theoretical predictions. The dominant instability mode was characterized by the cross-section profile of the root mean square of the velocity perturbations, which matches reasonable well with the eigenfunction of the most unstable eigenvalue. Also, the wavenumber was determined by correlating the velocity measurements in two sequential locations along the channel, which compares well with the theoretical value.