The strong contenders for being the rarest gemstone in the world are taafeite, musgravite and serendibite. According to the Gemological Institute of America, the so-called "rarity" of gemstones is not clearly defined, which means that determining the rarity of gemstones can be subjective.
Without a doubt, gemstones are considered as such because they are rare. Apart from being hard to find or occurring only in one or two locations, a few gemstones are scarce in the sense that only a few of them are known to exist. Among these really scarce gemstones are taafeite, musgravite and serendibite.
Taafeite was named after an Irish gemologist named Richard Taaffe, who discovered it in 1945. Its color ranges from red to purple, with the former being the rarest. If one were to combine all specimens of taafeite in the world, the collection will only fill about half a cup. Experts estimate that diamonds are a million times easier to find than taafeite.
Musgravite is similar to taafeite, so similar in fact, that some believe that some taafeite may have been misidentified as musgravite. This might make musgravite even rarer, since as of November, 2014, there are only eight known specimens of it in the world.
Serendibite is only known to occur in Sri Lanka, where it was discovered by D.P. Gunasekera. Only three faceted specimens of this gemstones are known to exist as of November, 2014.