I think it probably ended up in the Shire's Mathom house, because Frodo especially wouldn't have had a need for it in Valinor, since he wouldn't have had to bother about the Final Battle, being a small, adorable Hobbit.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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