Captain America's shield is composed of an alloy of Adamantium-Vibranium & Thor's Mjölnir is forged with Enchanted Uru.

Captain America's shield

Thor's Mjölnir

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    To be pedantic, the phrase “more indestructible” doesn’t mean much. If something is indestructible, it can’t be destroyed. It’s a bit like asking whether time or space is more infinite. – Paul D. Waite Dec 29 '13 at 12:23
  • I think what starbreaker wants to ask is: can the hammer destroy the shield or will it get destroyed in the due process... – Shwetabh Shekhar Dec 29 '13 at 12:32
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    @ShwetabhShekhar - if so, why not just ask it outright? I'm VTC'ing this but Starbreaker can of course edit it to make it better/clearer - right now it's too close to a "Gorilla vs Shark" question. – user8719 Dec 29 '13 at 12:33
  • @jimmyshelter 2 i agree with you but you must not forget he is new! He'll learn shortly... – Shwetabh Shekhar Dec 29 '13 at 12:40
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    Starbreaker, don't repost questions that have been closed. If the linked duplicate doesn't address your issue, edit into your question an explanation for how your question differs from the duplicate. – user1027 Dec 29 '13 at 19:19

In the Marvel Canon, neither Mjölnir or Captain America's shield are "indestructible", as evidenced by their repeated destruction.

Captain America's shield, which is an Adamantium-Vibranium alloy has been broken five times in the history of comics. The Molecule Man once destroyed the shield (along with Mjolnir, Namor's indestructible armor, and the Silver Surfer's board) by unmaking them on the molecular level. Klaw and Dr. Doom (both amped by the power of The Beyonder) damaged the shield on separate occasions. Thanos, wielding the Infinity Gauntlet, effortlessly shattered the shield with a backhand.

King Thor's Eyebeams cut through Captain America's shield in the Alternate Universe comic "All Star Superman Vs Odin Force Thor".

The Destroyer cut Mjolnir in half in 'Mighty Thor Vs The Destroyer'

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