This paper discusses an unsteady separated stagnation-point flow of a viscous fluid over a flat plate covering the complete range of the unsteadiness parameter *β* in combination with the flow strength parameter *a* (>0). Here, *β* varies from zero, Hiemenz's steady stagnation-point flow, to large *β*-limit, for which the governing boundary layer equation reduces to an approximate one in which the convective inertial effects are negligible. An important finding of this study is that the governing boundary layer equation conceives an analytic solution for the specific relation *β* = 2a. It is found that for a given value of $\beta \u2009(\u22650)$ the present flow problem always provides a unique attached flow solution (AFS), whereas for a negative value of *β* the self-similar boundary layer solution may or may not exist that depends completely on the values of *a* and *β* (<0). If the solution exists, it may either be unique or dual or multiple in nature. According to the characteristic features of these solutions, they have been categorized into two classes—one which is AFS and the other is reverse flow solution (RFS). Another interesting finding of this analysis is the asymptotic solution which is more practical than the numerical solutions for large values of *β* (>0) depending upon the values of *a*. A novel result which arises from the pressure distribution is that for a positive value of *β* the pressure is nonmonotonic along the stagnation-point streamline as there is a pressure minimum which moves toward the stagnation-point with an increasing value of *β* > 0.