This paper proposes a new family of discontinuous PWM strategies to control the quasi-Z-source nine-switch inverters (qZS-NSI). The presented strategies provide buck and boost inversion capabilities, and suitable for common-frequency and different-frequency modes of operation. Accordingly, two different shoot-through (ST) approaches are introduced and compared. The first approach uses three-leg ST, while the second uses single-leg ST to reduce the number of switching commutations, therefore minimizing switching losses. Both approaches can be implemented using simple-boost (SB) and maximum boost (MB) control methods. The operating principles, performance criteria, and PWM modulator of each scheme are introduced. Compared to the conventional PWM technique for the same output voltage gains, the proposed schemes ensure continuous input current with minimum ripples, and the voltage stresses on the switching devices and capacitors could be reduced in the proposed MB control schemes. Moreover, the effective switching frequency of upper and lower switches of all schemes is fixed and could be reduced by 1/3 from the switching frequency of the conventional technique of the qZS-NSI, while only the single-leg ST schemes ensure minimum effective switching frequency of the middle switches. The proposed modulation strategies are digitally implemented and tested on the LAUNCHXL-F28379D DSP. The feasibility of the proposed modulation schemes is confirmed via simulation and experimental results, which show good agreement with the theoretical analysis. Moreover, the presented strategies can be applied to other types of Z-source NSIs.