First off, each Duelist has several Decks, like in the Tag Force Games

• Yuma is seen playing a Deck that uses “Number 39: Utopia“.
• Shark is seen playing a Deck that uses “Bahamut Shark“, “Double Shark” and “Xyz Remora“.
• Kaito/Kite is seen playing a Deck that uses “Galaxy-Eyes Photon Dragon“.
• Astral is seen playing a Deck that uses “Number 39: Utopia” and “Number C39: Utopia Ray“.
• Kotori/Tori is seen playing a Deck that uses “Little Fairy“, “The Agent of Miracles – Jupiter“, “Archlord Kristya” and “Fairy Cheer Girl“
• Tetsuo/Bronk Stone is seen playing a Deck that uses “Tin Goldfish” and “Tin Archduke“.
• Class Rep Todoriki/Caswell is seen playing a Deck that uses “Crashbug X“, “Crashbug Y” and “Crashbug Z” and the game exclusive “Antidote Nurse“.
• Tokunosuke/Flip is seen playing a Deck that uses “Chewbone“.
• Cathy is seen playing a Deck that uses “Cat Girl“, “Stray Cat Girl” and “Cat Girl Magician“.
• Orbital 7 is seen playing a Deck that uses “Super Defense Robot Monki“, “Super Defense Robot Elephan” and “Googly-Eyes Drum Dragon“.
• III/Trey is seen playing a Deck that uses “Number 33: Chronomaly Machu Mech“
• IV/Quatro is seen playing a Deck that uses “Number 15: Gimmick Puppet Giant Grinder“.
• Anna is seen playing a Deck that uses “Snow Plow Hustle Rustle“, “Night Express Knight” and “Superdreadnought Rail Cannon Gustav Max“.
• Sanagi Chono/Scarlett Starling is mentioned to be playing a “Madolche” Deck.

The first three characters whose story modes you can pick are Yuma, Shark and Kaito’s.

Source:
http://www.konami.jp/gs/game/yugioh_dc/duelist
http://blog.livedoor.jp/maxut/archives/34039570.html