Choose one or two for each so others can post scenes not yet mentioned.
I will steal the most obvious...
Best Battle Scenes: Saving Private Ryan (Omaha Beach Landing)
Return of the King (battle for Minas Tirith)
Best Fight Scenes: Star Wars Ep. 2 (Yoda vs. Count Dooku)
Legend of the Drunken Master (Axe gang scene)
I will steal the most obvious...
Best Battle Scenes: Saving Private Ryan (Omaha Beach Landing)
Return of the King (battle for Minas Tirith)
Best Fight Scenes: Star Wars Ep. 2 (Yoda vs. Count Dooku)
Legend of the Drunken Master (Axe gang scene)