Most Rare Gemstones in the World rear than a Diamond

Pianite

In 2005, The Guinness Book of World Records called painite the world's rarest gemstone mineral. First discovered in Myanmar by British mineralogist Arthur C. D. Pain in the 1950s, for decades there were only two known crystals of the hexagonal mineral on Earth; by 2005, there were still fewer than 25 known specimens.
Colors may vary from brown to red and pink. It is also pleochroic, meaning it can emit different hues depending on the angle you are coming from. When placed under short wave ultraviolet light, this rare stone will emit and fluoresce a strong green color.

Black Opal is a greenish type of opal with black mottling and gold flecks. Usually found in ancient hot springs, the gem is usually tumbled smooth and cut cabochon.The Black Opal differs from other Opal as it is formed on a darker (black) quartz-like layer that enables greater refraction/reflection of the light to the top of the opal, especially the reds and pinks.
Up to 20.000 $ per carat
Musgravite

Named after the Musgrave Ranges, Australia, where it was first discovered. Its hardness is 8 to 8.5 on the Mohs scale. Only eight specimens of this gem have been identified. Musgravite, however, is very similar to another gem called taaffeite, so it is possible that some taaffeites have just been misidentified. Up to $35.000 per carat.
Alexandrite

Alexandrite is a genuinely incredible gemstone, The alexandrite variety displays a color change (alexandrite effect) dependent upon the nature of ambient lighting. Alexandrite effect is the phenomenon of an observed color change from greenish to reddish with a change in source illumination. To be clear: this color change is independent of your viewing angle; a gemstone that shifts colors when you rotate it in your hand, is said to be pleochroic, and while alexandriteis strongly pleochroic, it can also change colors independently of viewing angle when viewed under an artificial light source.A variety of Chrysoberyl, alexandrite belongs to the same family of gemstones as emerald. up to 100.000$ per carat.
Poudretteite

Poudretteite is an extremely rare mineral and gemstone that was first discovered as minute crystals in Mont St. Hilaire, Quebec,Canada, during the 1960s. It was initially thought to be extremely rare, but several of this gemstone was discovered in Myanmar in 2003.