Based on the US SNES version of the game. All footage was captured by myself using ZSNES v1.51. Special thanks to Jackie's Perfect Game Saves and The_Fake_God's Final Fantasy III Save State Editor.

Items 00:00
White Magic 01:30
Black Magic 02:29
Gray Magic 04:57
Espers 07:01
Cyan's SwdTechs 10:54
Shadow's Thrown Items* 11:40
Edgar's Tools 12:11
Sabin's Blitzes 12:49
Strago's Lore 13:41
Setzer's Slots† 16:11
Mog's Dances 16:46
Gau's Rage Abilities‡ 19:33
Umaro's Storm 23:36
MagiTek 23:42
Desperation Attacks 24:23

* Includes only Shadow's special throwing items (three ninja stars and five scrolls).
† Excludes the 3Bar and Bahamut combinations, since they're identical to the regular Esper attacks.
‡ Excludes regular magic, Strago's Lore and Mog's Dance attacks.

Let me know if I missed anything.