The two-dimensional Navier-Stokes equations for a Newtonian fluid in the absence of body forces are considered in a rectangular double-lid-driven cavity with free surface side walls, the cavity aspect ratio A and three cases of the ratio (S = 0, -1, 1) of the upper to the lower lid speed. Using a finite element formulation with a mesh which is adaptively refined to facilitate the location of stagnation points, the effect of Reynolds numbers (Re) in the range [0, 100] on the streamline patterns and their bifurcations is investigated as A is varied for each S. For Re --> 0 and each S as A is decreased, a sequence of pitchfork bifurcations at a stagnation point on x = 0 is identified as seen in the work of Gaskell et al. [Proc Instn Mech Engrs Sci Part C 212 (1998) 387]. As Re increases for S = 0 and decreasing A the stagnation point on x = 0 disappears and away from x = 0 cusp (saddle-node) bifurcations arise rather than the pitchfork bifurcation whereas for S = -1 and Re is an element of [0,100] the origin is always a stagnation point at which the same type of bifurcations arises. (C) 2002 Elsevier Science Ltd. All rights reserved.