This paper addresses the problems of the input-to-state stochastic stability for neutral switched stochastic delay systems. Two switching signals in the input controller, including the external input disturbance, are considered: (1) the synchronous switching signal, which indicates that the switching signal of the input controller coincides with that of the controlled subsystems, and (2) the asynchronous switching signal, which signifies that two switching instants for the input controller and the subsystem are not identical. Irrespective of which switching signal exists in the input controller and the subsystem, multiple Lyapunov-Krasovskii functions, generalized delay integral inequality, and mode-dependent average dwell time are incorporated to analyze the underlying problem. Sufficient conditions based on two integral inequalities are obtained for these two cases, respectively. Also new and more relaxed Lyapunov monotonicity conditions are introduced. One simulation example is provided to demonstrate the effectiveness of the theoretical results.