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Summary: Fifth‐generation (5G) networks are expected to offer unprecedented amounts of spectral and energy efficiency. Spatial modulation (SM) techniques have the capability to achieve an appealing trade‐off between these conflicting design parameters. Fully generalized SM (FGSM) and fully quadrature SM (FQSM) are the recent high‐rate SM variants, where the spectral efficiency is linearly proportional to the number of transmit antennas. The transmit antenna selection schemes can efficiently improve the average bit error rate (ABER) performance of SM techniques. The main objective of this work is to investigate the employment of transmit antenna selection schemes to FGSM and FQSM. Initially, Euclidean distance optimized antenna selection (EDAS) scheme is employed to FGSM and FQSM. It offers superior performance than conventional FGSM/FQSM without transmit antenna selection at the cost of higher computational complexity. In order to reduce the complexity cost, four suboptimal schemes based on channel capacity, correlation and combination of them are proposed and employed to FGSM and FQSM. The suboptimal transmit antenna selection based on capacity and correlation (TAS‐A‐C) offers a minimum gain of ∼6 dB over conventional FGSM/FQSM with significantly lower complexity. In addition, the performance of all schemes is investigated for increased number of available transmit antennas. As the proposed schemes address the spectral and energy efficiency trade‐off effectively, while corroborating better ABER performance, they have the potential to become a competing candidate for next‐generation networks. [ABSTRACT FROM AUTHOR]