Nearly all of the research in this search says there is either no statistical difference in effectiveness or “The diminished force output suggests that the overload stresses required for strength training necessitate the inclusion of resistance training on stable surfaces.”
Training on unstable surfaces is very limited in the real world applications. How often do you find yourself on an unstable surface while moving a load other than yourself? Maybe carrying objects on a loose stone/sand surface? There is application of stability training for fitness. But it does not involve getting on an unstable surface. While using a bosu (and never using the deck,that is just an accident waiting to happen) or other inflatable or balance board etc. can help to increase reactive balance and stabiity, it is not wise to use such equipment under any actual load. (As an extreme example, loaded squats on a bosu. Why?)
I teach CEC courses called “Stability Resistance Training” that I like to think is very cutting edge and informative. So called stability equipment is discussed and the limitations/potential safe uses are covered. But the bulk of the courses cover training the stability system in a very sound science based program. As in all forms of exercise; proper initiation, progression, and regression cannot be compromised. Unfortunately, these considerations are neglected by a large number of instructors and exercisers.
Interested instructors can contact me through my profile or website, www.hawaiifitnessacademy.com .