We analyse the gravity driven flow of ice down the slopes of Antarctica. Ice streams are almost invariably observed, and these could be explained by a transverse instability of the gravity driven flow of ice. In Newtonian fluids, the longitudinal instability is guaranteed to be the predominant one, and hence any transverse instability, if possible, must be due to the non-Newtonian character of ice. We show that the second-order Coleman-Noll fluid, and the lower-convected Maxwell fluid, could indeed show a predominant instability in the transverse direction, and we advance two speculations: first, that the predominant instability is either the longitudinal or the transverse one, and second, that the latter may be the case only for fluids which exhibit a non-zero second normal stress difference in shear flow. We also show that, for the specific problem at hand, the second-order constitutive equation is plausible.