Fairy Tail has three legendary magic: Fairy Law, Fairy Glitter, and Fairy Sphere.
Can anyone outside of Fairy Tail learn and use these magic?
Yes and no, some of the Fairy spells are distinctive to the guild, and the skills that can be used by outsiders are not easily learned/taught. This makes the likelihood of Fairy spells being used outside of Fairy Tail very unlikely.
As for leaving members who know one or more of the 3 great spells, there is a rule which states:
1. You must never reveal sensitive information about Fairy Tail to others for as long as you live.
Fairy Sphere, cannot be used by outsiders
Fairy Sphere uses the Fairy Tail member mark (pair of wings mark) and the emotion of those who wield it to obtain magical power. Therefore, this spell can not be used by outsiders.
Fairy Glitter, cannot be used by outsiders
As it is a distinct spell of Fairy Tail, and the fact that it is sealed in the grave of Mavis and needs the approval of Mavis herself to use, it makes it highly unlikely that anybody outside of the guild members can/will ever be allowed to use this.
Fairy Law, most likely can be used by outsiders
It is also a distinct spell of Fairy Tail, but most likely, it can be used by outsiders.
We know for a fact that there has been at least one instance where an outsider knew how to use Fairy Law, since Laxus also knew how to use this skill. Another case is Precht, the second guild master of Fairy Tail, now known as Hades, who left the guild but could still use Fairy Law.
The likelihood of learning this skill outside of Fairy Tail is close to 0, making it not likely to ever happen.
Any magic can be used by anyone, Fairy Sphere included. Mavis only required the guild members' emotions because she was a spirit and had no magical energy. All Three Grand Fairy Magics are simply uber-powerful light-based magic.
Not one part of the manga or anime does it say any form of magic is exclusive. The Three Grand Fairy Spells are simply based around light magic. Mavis learnt Law from Zeref in Fairy Tail Zero, used it too early and was cursed. She then altered and adjusted it (it was previously black magic) and created Fairy Law.
Zeref taught Mavis Law, so there's also a possibility that he is able to use any of the three spells. Mavis's main magic is light-based, including her illusion magic. Mavis said that she used the bonds and emotions of the guild and converted it into magic to cast fairy spells.
After that, it is clearly a light spell. Hence the reason when the spell was released, a butt tonne of light erupted from the water. With Fairy Glitter collects and concentrates the light of the sun, the moon, and the stars.
Fairy Law converts the user's magic into light that envelops everything. Hades had his own version that uses darkness as a countermeasure for Fairy Law since it's light.
I don't know about the other two, but Fairy Law is actually based on the spell "Law" which Mavis used in Fairy Tail Zero (the story about how Fairy Tail was founded).
This is shown for the first time in chapter 10. Mavis learned "Law" from Zeref, so if we say that "Law" and "Fairy Law" are the same, people who are not in Fairy Tail can also learn this Spell (Zeref for example). If "Fairy Law" is a development of "Law" created by Mavis, probably only members of her guild can use this spell because no member of Fairy Tail would tell this secret to an outsider. "Law" on the other hand can be learned by people who are not in Fairy Tail too. The problem here is, that it takes ten years to master this spell (according to Zeref at least).