From the Super Mario Wiki
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“AY-O! Did ya see that? Those Prankster Comets were green! That means there are Green Stars around now! Those must be the cosmic jewels I keep hearing about! Those Green Stars are pretty hard to find, so keep your eyes and ears peeled! Take it away, Captain!”
Super Mario series
Super Mario Galaxy
In Super Mario Galaxy, there are only three Green Stars, each of which is hidden away in a different galaxy. Collecting all three of them grants Mario access to the three Trial Galaxies on the Planet of Trials: the Rolling Gizmo Galaxy, the Loopdeeswoop Galaxy, and the Bubble Blast Galaxy.
Super Mario Galaxy 2
In Super Mario Galaxy 2, there are a total of 120 Green Stars to be found. These are only available once the player collects 120 regular Power Stars and then completes the first mission of Bowser's Galaxy Generator again. Each galaxy has the same number of Green Stars as regular Power Stars (except the Grandmaster Galaxy, which has no Green Stars at all). When collecting Green Stars, the player will have to do a certain mission of a galaxy again. In some cases the Green Stars in a galaxy can be collected in any order (e.g. "Green Star 2" can be collected during the Green Star 1 mission), but in general they can only be collected when playing a Green Star mission. Green Stars are usually found in hard to reach or hidden areas of the galaxies. Additionally, the Green Stars only appear when a galaxy is orbited by a Green Comet, which will appear once they are unlocked.
Super Mario Galaxy 2's Green Stars make a distinct twinkling sound, which is different from that of normal Power Stars and the Green Stars in Super Mario Galaxy and is meant to help players in locating them.
Super Mario 3D World
In Super Mario 3D World, Green Stars function similarly to Star Medals from Super Mario 3D Land and share the same sound effects. Unlike Star Medals, there are courses where there are more than three of them at a time. They appear in sets of one in Enemy Blockades, three in main courses, five in The Adventures of Captain Toad stages, or five, ten, and thirty in Mystery Houses. In Enemy Blockades the player needs to defeat all of the enemies to receive the Green Star. In main courses they are scattered around in hidden areas just like Star Medals. In the Adventures of Captain Toad levels the objective is to solve the course's puzzles to obtain all five Green Stars. In Mystery Houses, the player needs to complete one of the ten second challenges to receive one. There are 380 Green Stars in total and each of them additionally grants the player 4000 points. Some levels on the World Map require the player to have a certain number of Green Stars to play the level and advance past it.
Captain Toad: Treasure Tracker
Green Stars also appear in Captain Toad: Treasure Tracker. After the end credits, a Green Star is falling across the sky that Captain Toad chases after. The star falls down the Clear Pipe leading to the Sprixie Kingdom from Super Mario 3D World which Captain Toad also enters. Green Stars also appear in place of Power Stars at the end of the Super Mario 3D World levels in the Bonus episode.