Calf muscles
They're the muscles most associated with jumping, and they play the greatest role. Without your calves, you would have a lot of trouble getting more than a few inches off the ground. These muscles are worked out by jumping rope and toe raises, and the stronger and more explosive they are, the more your vertical jump will benefit.
Hamstring muscles
While the calves are the most explosive muscles used in your jumping mechanics, the hamstring is the most powerful. Your hamstrings also play a crucial role in balance and stability as you project your body upward. As a result, many training exercises and lifts focus on the hamstring muscles' ability to stabilize, rather than explode upward.
Tibialis anterior
This muscle, located on the shin, provides a burst of quickness and power as you jump upward. It is activated by your toes when you jump. Developing this muscle can not only increase your vertical leap, it should also reduce your risk of shin splints.
Buttocks
Your butt muscles act as a pivot point between your lower and upper bodies. The stronger they are, the quicker they can snap your two halves in alignment with one another and transfer energy upward into your upper body.
Back muscles
Your back muscles, particularly the lower back, have to be strong to maintain their position as you crouch down and project your body upward. Your muscles also have to be strong enough to arch toward the sky and obtain your maximum leap.
Abdomen
Without your abdomen muscles holding tight, your lower body would get stuffed by your slumping upper body as you try to jump.
Shoulders and arm
As you jump upward, the power generated from swinging your shoulders and arms upward can provide a little more force going against gravity and could create extra space between you and the ground.