Solid-state transformers (SSTs) are expected to become one of the most powerful and adaptable devices that allow controllable voltage, power factor correction, fault isolation, compact size as compared to their low-frequency (50 Hz/60 Hz) counterparts. The high-frequency isolation of single-phase modified quasi-Z-source AC–AC converter (SPM-qZAC) employing bidirectional switches to create the single-stage SPM-qZAC-based SST is proposed in this paper. The proposed topology offers all the benefits of conventional impedance source topologies, including single-stage power conversion with a small footprint, buck–boost operation and retaining or reversing the phase angle. Moreover, the presented converter topology allows to share the same ground between input and output voltage, continuous input current, no input–output LC filters and performs AC–AC power conversion without the use of DC storage, making it suitable for AC voltage regulation. This study introduces two modulation schemes for SPM-qZAC-based SST. Finite control set model predictive control (FCS-MPC) which is a current control technique at variable switching frequency is employed owing to the capabilities of modern digital signal processing. Further, an adaptive hysteresis band-based delta sigma modulation (DSM) technique that offers the benefit of constant switching frequency (CSF) is proposed as an alternative voltage control-based modulation of the same topology. Various performance indices including steady-state response, total harmonic distortion of source current and dynamic response are assessed through simulation studies using MATLAB/Simulink software and real-time simulation environment using RT-Lab with OPAL-RT OP4510. It is observed that SPM-qZAC exhibits good performance when modulated using either modulation techniques; however, CSF-DSM technique offers an additional benefit of constant switching frequency.