@KingOfBigOz is right that, for most of the scenarios, there isn't a single "best" Civ - rather, each Civ has certain advantages and disadvantages in the scope of the scenario. However, here are my thoughts:
Conquest of the New World - Really any Civ; Spain's probably the easiest (as they get a starting caravel out in the middle of the Atlantic), but they're all pretty fun and none are too tricky.
Paradise Found - Again, really any of the Civs can be the "best". The scenario's just not all that much fun after the first time you've played it, though.
1066: Year of Viking Dynasty - Norway's probably the easiest, in all honesty, while Anglo-Saxon England's the hardest (but can be the most fun, too.)
Samurai Invasion of Korea - All fun. Japan's the easiest, Korea's probably the hardest.
Wonders of the Ancient World - Since you can play on a randomized map, it's hard to point to a Civ and say "this is the best" based on where they start. As for their UAs, it completely depends on how you want to play the scenario.
Into the Renaissance - Russia's probably the hardest because they get the fun of a Mongol invasion part of the way through the game. Any of the other Civs are the most fun.
Fall of Rome - Either part of the Roman Empire is the hardest (though the Western Empire is more difficult), while any of the Barbarians are the easiest.
Empires of the Smoky Skies - Like with the WotAW Scenario, the "best" Civ depends entirely on how you want to play.
Scramble for Africa - England's the easiest, The Zulu are the hardest, all Civs are fun.
American Civil War - There are two Civs. They're both easy. The scenario in general is dull.