Using inverse methods constrained by recent satellite observations, we have produced a comprehensive estimate of the basal shear stress beneath the Filchner-Ronne ice streams. The inversions indicate that a weak bed (approx. 4-20kPa) underlies much of these ice streams. Compared to the Ross ice streams, the distribution of weak subglacial till is more heterogeneous, with 'sticky spots' providing much of the resistance to flow. A weak bed beneath Recovery ice stream extends several hundred kilometres inland with flow. Along this ice stream, discrepancies between thickness measurements and flux estimates suggest the existence of a deep (-1400m) trough not resolved by existing maps of subglacial topography. We hypothesize that the presence of this and other deep troughs is a major influence on this sector of the ice sheet that is not fully incorporated in current models of ice-sheet evolution.