Emerald Shards (エメラルドのかけら,Emerarudo no kake-ra?, lit. Emerald Fragments) are fragments of either Chaos Emeralds or the Master Emerald crushed by an external force that could fracture them into a variety of fragments that can be merged together to reform the emerald, by putting them in one place.
In the Dreamcast version of the game, they are a very dark green and hard to see against some backgrounds, but in Sonic Adventure DX: Director's Cut, they are much more radiant and easier to see. When touching an shard, it will float up over Knuckles and disappear in a burst of sparkles.
Sonic Adventure 2
In Sonic Adventure 2 and its later iteration Sonic Adventure 2: Battle, Rouge and Knuckles were fighting over the Master Emerald. Dr. Eggman attempted to steal it, but Knuckles smashed it to prevent the theft, so he and Rouge began hunting down its Emerald Shards. In the majority of Knuckles and Rouge's Stages, the objective is to find all the Emerald Shards' present. Once again, the player has the Emerald Radar to pinpoint their locations. In the Treasure Hunt Races in the game's two-player mode, the Emeralds Shards are to be located by the players in a similar fashion.
The two treasure hunters eventually gathered all the shards, though after Knuckles saved Rouge's life, she returned her shards to Knuckles with reluctant gratitude, which let Knuckles reassemble the Master Emerald.
In Sonic Battle, Emerald Shards from both the Master Emerald and the Chaos Emeralds appeared. Many of the Chaos Emeralds were crushed by Dr. Eggman into five Emerald Shards and used to operate the E-121 Phi model robots. Once a Phi is destroyed, they drop their Emerald Shard and either break down or retreat. When five Emerald Shards of the same color are united, a Chaos Emerald can be created. Tails is the one who first discovered how Emerald Shards work and characters often visit him to have their Chaos Emerald reassembled. Chaos Gamma is also seen with an Emerald Shard which he absorbs to injure Shadow.
When Sonic tried to use the Master Emerald to neutralize the Chaos Emeralds inside of the out-of-control Emerl, Emerl smashed the Master Emerald into tiny pieces and the pieces were not seen for the remainder of the game, though it is highly likely they were recovered and reassembled. At the end of Emerl's story, Emerl exploded, smashing the Chaos Emeralds within him, turning them to Shards with Emerl's "soul" within them.
In Sonic Runners, the Emerald Shards appeared again as fragments of the Master Emerald called Master Emerald Fragments. When Dr. Eggman tried stealing the Master Emerald, Knuckles smashed it into several fragments which dispersed far and wide to keep it safe. Both Team Sonic and Eggman thus began tracking down the fragments. While Team Sonic found some without trouble, they sometimes had to fight Eggman and Metal Sonic for them. Eggman initially had the upper hand thanks to an Emerald Detector, but Team Sonic took it from him and use it to find the remaining shards and reassembled the Master Emerald.
In other media
The Emerald Shards make a few appearances in the anime series Sonic X. In this media, the only Emerald Shards shown are the fragments of the Master Emerald. Knuckles himself is shown to have limited control over the Emerald Shards which lets him levitate them through midair.
The Emeralds Shards first appeared during the Chaos Saga, where Eggman shattered the Master Emerald to free Chaos. As such, Knuckles went off on a search across the nearby areas on earth for the Emerald Shards to restore the Master Emerald. He eventually got all the Emerald Shards back and managed to rebuild the Master Emerald. The Emerald Shards appeared again near the end of the Metarex Saga where the Master Emerald shattered into several Emerald Shard after all its power was used to defeat Dark Oak. Knuckles then brought the shards back home to Angel Island where he could watch over them until their powers would return.
Strangely, Tails claimed that he could reassemble the Emeralds if the Shards were of the same color. However, all the Chaos Emeralds in Sonic Battle were given the same green Chaos Emerald sprite.