We report on a three-dimensional MHD numerical experiment of a small-scale coronal mass ejection (CME)-like eruption propagating though a nonmagnetized solar atmosphere. We find that the Kelvin-Helmholtz instability (KHI) develops at various but specific locations at the boundary layer between the erupting field and the background atmosphere, depending on the relative angle between the velocity and magnetic field. KHI develops at the front and at two of the four sides of the eruption. KHI is suppressed at the other two sides of the eruption. We also find the development of Alfvénic vortex shedding flows at the wake of the developing CME due to the 3D geometry of the field. Forward modeling reveals that the observational detectability of the KHI in solar eruptions is confined to a narrow ≈10 range when observing off-limb, and therefore its occurrence could be underestimated due to projection effects. The new findings can have significant implications for observations, for heating, and for particle acceleration by turbulence from flow-driven instabilities associated with solar eruptions of all scales.