Trances

Zidane

Free Energy Deals physical damage to the enemy Learned from Flee
Tidal Flame Deals physical damage to all enemies Learned from Detect
Scoop Art Deals physical damage to the enemy Learned from What's That!?
Shift Break Deals physical damage to all enemies Learned from Soul Blade
Stellar Circle 5 Deals physical damage to the enemy Learned from Annoy
Meo Twister Deals physical damage to all enemies Learned from Sacrifice
Solution 9 Deals physical damage to the enemy earned from Lucky Seven
Grand Lethal Deals physical damage to all enemies Learned from Thievery

Steiner -- Fight
When Steiner undergoes Trance, the damage he deals increases two fold. When Tranced, he can be the most valuable asset to your party considering his Trance costs no MP.

Garnet AKA Dagger -- Eidolon
This one really isn't too special. It simply allows for a better chance of a full sequence of animation for an Eidolon. This will increase its attack power somewhat but this same thing can easily be done by equipping the support ability, Boost.

Vivi -- Double Black
When Vivi Trances, he will be allowed to cast two black magic spells in a row against the enemy party. This can result in some of the most powerful damage towards the end of the game.

Freya -- Jump
Freya's Trance can be useful in near death situations. After a Trance has been undergone, select the Jump command. Freya will launch herself out of harms reach and stay there for a while, periodically blasting the enemy with her lance. The damage varies though.

Quina -- Cook
This only comes in handy when you are attempting to learn Blue Magic, which is gained through eating monsters. Quina's Eat command will turn into the Cook command. This allows you to eat an enemy at 50% of its HP rather than having to do the usual 25%.

Eiko -- Double White
Eiko's Double White Trance ability can come in really handy, especially when your HP is low and you still want to attack the enemy. Double White allows you to cast two white magic spells on your allies or enemies. Doing a double of Curaga and Holy when the need arises, could be handy.

Amarant -- Elan
Amarant's Elan Trance ability is an evolved version of the Flair command. It allows all Flair abilities to be used on all enemies or all allies.