Abstract The simultaneous effects of normal transpiration through and tangential movement of a semi-infinite plate on self-similar boundary layer flow beneath a uniform free stream is considered. The flow is therefore governed by a plate velocity parameter λ and a transpiration parameter μ and the computed wall shear stress parameter is f″(0). Dual solutions are found for each value of μ in λ– f″(0) parameter space. It is shown that the range of known dual solutions for zero transpiration increases with suction and decreases with blowing. A stability analysis for this self-similar flow reveals that, for each value of μ, lower solution branches are unstable while upper solution branches are stable.