The yield conditions for the displacement of three-dimensional fluid droplets from solid boundaries are studied through a series of numerical computations. The study considers low-Reynolds-number shear flows over plane boundaries and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio γ, capillary number Ca and advancing and receding contact angles, θA and θR. This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(γ, θA, Δθ) where Δθ=θA−θR is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on solid surfaces. Additional constraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While good agreement is found in the joint asymptotic limits Δθ[Lt ]θA[Lt ]1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets.