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This includes any move or item that can do something in a double battle that it can't in a single battle, such as helping hand, plus and minus, and The pledge moves.

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Two things that come to mind are Ally Swap and Pollen Puff’s affect when targeting an ally.
Plus and Minus also work in single battles with the move Magnetic Flux

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A universal circumstance:

  • Moves like Rock Slide that hit all opposing targets only deal 75% damage in a Double/Multi Battle, or 50% in Gen III.

Moves that only work in Double Battles:

  • Helping Hand
  • After You
  • Quash
  • Helping Hand
  • Ally Switch
  • Follow Me
  • Rage Powder
  • Aromatic Mist
  • Hold Hands
  • Spotlight

Moves that have special effects in Doubles:

  • Flame Burst: Takes away 1/16th HP of opponent adjacent to target
  • Acupressure: Can target the ally instead of the user
  • Pollen Puff: Heals ally instead of damaging them
  • Outrage, Petal Dance, Thrash, Uproar: Targets random opponent
  • And of course the long list of moves that hit multiple targets in a Double Battle. I won't list these, but you can view them in the source.

Abilities that have special effects in Doubles:

  • Lightning Rod: Redirects single-target Electric type moves
  • Plus/Minus: Boosts Sp.Atk by 50% if ally has Plus/Minus
  • Storm Drain: Redirects single-target Water type moves

Abilities that only have effects in double battles:

  • Healer: 30% chance of healing ally's status condition every turn
  • Friend Guard: Reduces damage done to ally by 25%
  • Telepathy: Avoids any damage derived from ally
  • Battery: The BP of allies' Special moves are increased by 30%
  • Receiver/Power of Alchemy: If an ally faints, the Pokemon with Receiver/Power of Alchemy receives the ally's ability.
  • Power Spot: Base power of ally's moves is increased by 30%
  • Commander
  • Costar: Copies ally's stat changes upon entering battle.

I think that's everything, let me know if I forgot anything. Sorry for my bad previous answer.

Source

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edited by
Plus, minus, flower gift, victory star, aroma veil, sweet veil, storm drain, and lightning rod affect the user in single battles. Flower veil can affect the user if it's grass type. Many of the moves in your link can still hit the opponent in single battles, such as surf.