Never wear jewelry on the trampoline. Bracelets, watches, earrings and even rings could snag on the trampoline mat and cause injury. If you land on a piece of jewelry you could get hurt. To avoid jewelry-related injuries, remove it and anything that might be in your pockets before you get onto the in-ground trampoline.
Somersaults are fun, but can be dangerous. If the person doing the somersault doesn't have the training or even the flexibility to fully flip, head and neck injuries -- sometimes very serious can occur. If someone has proper training and there is someone there to "spot" them, a somersault may be permitted, but you really need to take that on a case by case basis.
If you have an above ground trampoline, everyone should use the trampoline ladder to get into and off of the trampoline. Do not jump off the trampoline. The ground is probably farther away than you imagine and let's face it -- it's hard and if you're jumping up then off the trampoline you have an awful lot of height behind the jump and can break your leg bones.
Check the mat before you jump to make sure it's dry and clean. A wet mat will be slippery and can lead to injuries. If the mat is dirty, the grit or if there are tree branches or debris can injure the individual on the trampoline and can damage the mat itself.
If you have an above ground trampoline, it must be put on level ground. Don't just place it in the yard without assuring the ground is completely level. If it isn't level and someone jumps off to one side of the trampoline it could easily flip over and lead to injury. You may even want to place your trampoline on a surface of sand or even wood chips to help level the surface.
Most trampoline and in-ground trampoline injuries occur when common sense has flown out the window. Put safety measures in place and adhere to them for the safety of everyone.