Multi-stage gas switches (MSGS) are widely used in megavolt pulsed power systems due to their advantages of high withstand voltage, fast establishing time and relatively high operating frequency. Multiple operation modes, which are induced by the different breakdown sequence of each stage, are found to affect the breakdown characteristics of the gas switch. Based on the theoretical analysis of working principle, the classification of the operation mode and the factors determining the operation mode of the MSGS are proposed. Methods for investigating the output waveform for MSGS are established by simulating lumped parameters and studying the breakdown time sequence of individual stages. In comparison with the intended mode of operation, the disadvantages of the unintended modes of operation for MSGS, including error-breakdown and non-triggering mode, are revealed. For unintended modes, the distortion of the output waveform, which determines the efficiency of energy transmission through the switch, and the increased delay time jitter, which determines the synchronisation of the gas switch, are further investigated. The possibility of non-trigger and its mechanism on surface flashovers induced by drifting electrons are also analysed. Based on this study, design principles and methods are proposed to ensure the intended operation of the multi-stage gas switch.