The bifurcation phenomenon in flow through a curved rectangular duct is investigated in this study. The non-linear equations of motion governing the steady, fully developed laminar flow of an incompressible generalized Newtonian fluid have been solved numerically. Extensive results have been generated in an effort to map the regions of multiple solution in the parameter space of Dean number, Dn, aspect ratio, γ, power-law index, n, and radius of curvature, r. For a Newtonian fluid (n = 1), at a fixed curvature (r = 100), the transition between a symmetric 2-cell and a symmetric 4-cell solution appears to follow a tilted cusp. The extent of the stable, symmetric 2-cell solution surface is critically influenced by the length scale γ. In the non-Newtonian case, at a fixed aspect ratio (γ = 1) and a fixed curvature (r = 100), the flow transition follows that of a fold catastrophe. The influence of the curvature is reasonably well accounted for in Dn. The bifurcation set determined in the Dn-γ space remains qualitatively the same at any value of n or r. These parameters merely shift and/or stretch the equilibrium surface determined by Dn and γ.