How does one register a Restrictive Covenant in England?

There is nothing you can do. If you are the registered proprietor of your home, and you have the freehold, you have as much ownership as you're going to get. Restrictive Covenants are agreements between landowners (usually neighbours) to burden the land of one with some right for the benefit of the other's land. For example A might make a restrictive covenant with B that B will not build anything on his land more than two levels - to stop B building anything that overlooks A's land. Restrictive Covenants cannot be used to protect your property against the government. The government can always compulsorily purchase your property if it wants to, or pass a statute giving itself the right to dig underneath your property.

The only way you can create a restrictive covenant is by making it a condition of the sale when you sell the house. That would only be binding upon the new owner, not third parties.

There is no law against anyone digging deep below your home, or entering any natural cave or disused mine tunnel which might be beneth it, except in that they must not unreasonable interfere with your enjoyment of your property (e.g. by causing subsidence, noise or vibration). The same applies in the opposite direction - you cannot legally prevent aircraft from flying over your home at normal flying heights, but you could seek compensation if 'buzzed' by very low level aircraft.

If you are talking about Fracking, there is nothing you can do. You only own a few feet below your property, not down to the centre of the earth.

UK