The instabilities of the shear layer beneath the free surface of high-speed liquid jets are investigated. Such instabilities will generate waves on liquid-metal jet targets, affecting adversely the target performance. The most unstable wave number and the spatial growth rate of perturbation are predicted with linear stability theories and are shown to agree fairly well with experimental data for water jets. The effects of fluid surface tension and streamline curvature on the instabilities are analyzed to evaluate the applicability of water data to liquid-metal curved jets. It is shown that the surface tension effects are negligible when the Weber number based on the shear layer thickness is greater than six, and also the streamline curvature effects are negligible when the radius of curvature is more than 30 times greater than the shear layer thickness.