Acoustofluidic micromanipulation is an important tool for biomedical research, where acoustic forces offer the ability to manipulate fluids, cells, and particles in a rapid, biocompatible, and contact-free manner. Of particular interest is the investigation of acoustically driven sharp edges, where high tip velocity magnitudes and strong acoustic potential gradients drive rapid motion. Whereas prior devices utilizing 2D sharp edges have demonstrated promise for micromanipulation activities, taking advantage of 3D structures has the potential to increase their performance and the range of manipulation activities. In this work, we investigate high-magnitude acoustic streaming fields in the vicinity of sharp-edged, sub-wavelength 3D microstructures. We numerically model and experimentally demonstrate this in fabricating parametrically configured 3D microstructures whose tip-angle and geometry influence acoustic streaming velocities and the complexity of streaming vortices, finding that the simulated and realized velocities and streaming patterns are both tunable and a function of microstructure shape. These sharp-edge interfaces hold promise for biomedical studies benefiting from precise and targeted micromanipulation.