Two staple pieces of equipment in my home gym are a BOSU Ball and an exercise ball. Both work for helping with core stabilization.
Core, butt and hamstring
Think of the bridge and start with your legs at a 90-degree angle, with the top of your shoulders, neck and head rested on the exercise ball. Try to keep your body flat, no dipping the bum down or arching. This position is hard enough to hold and you will feel everything engage.
To initiate the legs even more, bring one leg up out in front and hold. Be sure to alternate legs. Repeat 5–8 times per leg.
BOSU ski and skate
Stand on the flat side of the BOSU, which means you will activate the core as soon as you get on—this is great for balance as is.
Slowly start alternating the leg burn by pushing the BOSU down side to side. The deeper you go the harder it is—you can also increase or slow down your pace.
Place the exercise ball against the wall and stand on one leg. As in a ballet relevé, slowly raise the heel until you are putting all the weight on the ball of your foot. If single leg relevés are too tough—stand with both feet together and relevé at the same time while pushing the ball into the wall for balance and core stability.
Wall sit, squat or lunge back
Place the exercise ball against the wall and stand with it at your mid-back. Bend your knees 90 degrees.
Wall sit—hold the position for 30–60 seconds.
Wall squat—bend and straighten the legs putting some pressure on the ball as you go.
Bring a single leg back into a lunge and repeat— 8–10 times per side.