A novel microfluidic technique has been recently proposed to produce quasi-monodisperse collections of microbubbles with a controlled size. In this technique, a gaseous stream is injected through a T-junction into a microchannel transporting a liquid current. The gas adheres to a hydrophobic strip printed on the channel surface. When the gas and liquid flow rates are set appropriately, a gaseous rivulet flows over that strip. The rivulet breaks up downstream due to a capillary pearling instability, which leads to a quasi-monodisperse collection of microbubbles. Motivated by this application, we here analyze the stability of both gas and liquid rivulets coflowing with a current in a quadrangular microfluidic channel. The results essentially differ from those of cylindrical jets because the contact-line-anchorage condition affects fundamentally the rivulet's instability nature. The temporal stability analysis shows that the rivulet becomes unstable not only for (unperturbed) contact angles larger than 90 degrees (as can be expected) but also for values smaller than that angle. Interestingly enough, the maximum growth factor exhibits a non-monotonic dependence with respect to the Reynolds number (i.e., the viscosities). In fact, there are intervals of that parameter where the fluid system becomes unstable, while all the perturbations are damped outside that interval. The gaseous rivulet does not stabilize as the Reynolds number decreases, which means that it can be unstable even in the Stokes limit and for contact angles less than 90 degrees. In addition, the stability of a flowing liquid rivulet is not determined by its contact angle exclusively (as occurs in the static case), but by the Reynolds number as well. Liquid rivulets with contact angles less than 90 degrees can be unstable for sufficiently high Reynolds numbers. (C) 2014 Elsevier Ltd. All rights reserved.