A modal global stability analysis is addressed here for the boundary layer (BL) flow across a plate under the influence of wall-normal non-uniform suction and injection. For non-uniform wall-normal profiles of suction and injection, namely, two variations such as linear and quadratic variations are considered and compared the results with the uniform profile and the Blasius BL. The linearized governing stability equations are derived from the standard procedures, where the steady base flow comes from the solution of the Navier-Stokes (N-S) equation using the OpenFOAM software. The spectral method is employed to discretize the stability equations spatially. The governing stability equations are converted into an eigenvalue problem (EVP) with suitable boundary conditions in the flow and cross-flow directions and solved by the ARPACK package based on Arnoldi’s iterative algorithm and Krylov subspace size. The results reveal that suction leads to stabilize the flow whereas injection promotes it. Also, the flow becomes more stable for the BL flow with uniform suction than the non-uniform suction. However, it becomes unstable for the uniform injection and stable for the non-uniform profiles of injection at lower Reynolds number and intensity. The 2D spatial wavelet structures of perturbations are also obtained for different profiles of suction and injection to observe the spatial growth of perturbations and found that the wavepackets are developed at an early stage and vanish before it leaves the domain in the case of suction, while, maximum spatial growth of the wavepackets is found close to the outlet in a case of injection.