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Bone Shards are an item found in the Reign of Giants DLC. They are obtained by hammering Bones and Skeletons, by destroying Tumbleweeds, when attacking Hound Mounds, or as vomited gifts from friendly Catcoons. Bone Shards are used in several crafting recipes.
After all the natural Bones, Skeletons and Hound Mounds in the world have been destroyed, the only way to farm Bone Shards with a 100% drop is by hammering the player's own skeleton, which is generated when the player dies. To be able to hammer it, player needs to be resurrected, otherwise the game will end.
From Don't Starve Together, in A New Reign update, Bone Shards can be dropped after attempting to open the Loot Stash with the Deer Antler, causing Klaus and his two Gem Deer to spawn. Killing a No-Eyed Deer with Antler has 50% probability to drop one Bone Shard.
In the Shipwrecked DLC, two Bone Shards can be obtained by digging up X Marks the Spot structures, and can also be obtained by hammering Spoiled Fish. Using a Trawl Net also has a chance to yield Bone Shards in deeper parts of the Ocean.