The potential of faults to show earthquake-generating slip instabilities depends not only on the intrinsic frictional properties of the fault zone, but also on the elasticity of the surrounding material. A velocity-weakening fault is expected to show increasingly unstable frictional behavior with decreasing stiffness parallel to the shearing direction. The roughness of fault planes can cause slip in the shear direction to be accompanied by fault-normal movement, working against stiffness in the fault-normal direction. In this dataset, we report on laboratory friction experiments in which we systematically vary the stiffness surrounding the fault in both the shear-parallel and fault-normal directions, to investigate under which conditions slip instabilities can occur. The experiments were performed using synthetic quartz powder in a single-direct shear device. We report the frictional strength, the frictional sliding stability following the rate-and-state frictional framework, and statistics of the slip events. Reducing the shear-parallel stiffness causes the known transition from stable sliding to slow slip to fast stick-slip, whereas reducing the fault-normal stiffness causes stick-slip instabilities without transitional slow events, and reduced stiffness in both directions produces complex behavior.