There aren't any. Braces work by using wires to move your teeth. The orthodontist bends the wires in very specific ways to make sure that the movements gently move your teeth. Chewing gum can bend the wires and even break them- potentially damaging your teeth, gums and tongue.
Sadly there is none. The problem with chewing gum with braces is because it can stick to the braces and cause them to come off the tooth. It can also get stuck on the wires and cause them to get deformed.
No gum is best. Chewing gum is likely to cause damage to braces and the wires that are moving your teeth. Constant chewing forces can end up causing the wires to break or can dislodge the braces from the teeth. Either will slow down treatment and or cause damage to your cheeks and gums.
Chew at your risk. Any gum can be chewed with orthodontic braces, and it will very likely not break away the appliance from the tooth. The big problem is that the gum tends to distort the wires that are responsible for moving the teeth, and as such your treatment can be greatly extended or end in a compromised occlusion. Delay your gratification for the unveiling of a great smile!
None. Think about opening a tin can and the lid doesn't come all the way off. You then wiggle the lid back and forth until it snaps off. Same thing with braces.
Denture-safe gum. Gum that claims to be safe for dentures is probably your best bet but check with your dentist.