The split-step method (SSM) is widely used for numerical solution of nonlinear evolution equations, e.g., the generalized NLS: $i u_t + u_{xx} + (V(x)+2|u|^2)u = 0.$ We will show how SSM's instability is analyzed using a modified linearized NLS for a high-wavenumber numerical error. For example, for the SSM which uses the finite-difference solution of its dispersive step, this modified linearized NLS is: $i\psi_t + \delta\psi - \alpha \psi_{\chi\chi} + V(\epsilon\chi)\psi + 2|u_{\rm sol}(\epsilon\chi)|^2 (2\psi + \psi^*) =0,$ where $\psi$ is proportional to the numerical error. We will show how the instability differs for a stationary, moving, and oscillating background pulse. We will also highlight differences between instabilities of the finite-difference and Fourier implementations of the SSM, and then discuss how these results can be used to predict the instability of the integrating factor and exponential time differencing methods.