After just beating this game, I felt like rank-ordering the characters in terms of purely usefulness in battle.
So, my rankings are:

1st - Terra
I don't know if this is because I got her to the highest level out of all my characters, but she just seemed very useful and good with magic. Morph was good too

2nd - Sabin
Bum Rush FTW!

3rd - Edgar
I found his tool ability, particularly drill, very useful

4th - Gogo
Just because you could do blitz with him, and also all the magic you've ever learned

5th - Celes
Another character I got to a high level. Even though I didn't use Runic much, she was good with magic

6th - Setzer
Basically because of the fixed dice

7th - Relm
Even though I didn't use Sketch much, she was again another character good with magic

8th - Shadow
Average. Throw was ok but you had to re-stock it. Bonus points for the dog moves

9th - Mog
Dance could be useful, but it could also be bad

10th - Locke
He's an all-rounder, although there's nothing stand-out about him in battle IMO

11th - Gau
Hated at first because of lack of control. Became better when learning magic

12th - Strago
Just nothing to him. I didn't use him much, nor take him to Kefka's tower

13th - Cyan
His sword techniques really weren't for me. They took too long and weren't that useful IMO

14th - Umaro
Can't control him, so what's the point?

How would you rank the characters? (In terms of usefulness in battles)

1:Terra:She is strong with ultima and a good attacker.She gets ultima blade.There is another sword you can get in arena called illumina which increases evade i believe.
2:Celes:She has the absorb ability, can wear minerva, wear strong weapons like illumina and ultima sword, and casts magic.She is good at magic too.
3:Gau:I think gau is actually underrated.With a certain item, you can do the wind god trick, and with the right farming, you can have all his strongest abilities and deal very good damage even in the world of ruin(Cat scratch anyone?) And that's not all:He has access to very strong spells too.

GoGo:He does weaker damage when using unique abilities, but can use any spell others can, and he can use mimic to mimic bosses too.He is pretty good actually.

4:Shadow:Early in world of balance, he was good but he gets stronger in the world of ruin because of access to arena and strong throw weapons.His dog is quite damaging too.
5:Sabim:Bum rush is great, and he gets access to a lot of abilities which are nice.His best ability is single attack Bum rush, because it always deals 9999 damage.His healing ability is very nice too.I think he shines most though in the world of balance, because damage is relatively low, and he deals more damage than anyone else can early on.

5:Mog:His abilities can range from very useful, to completely useless in the world of ruin.In the world of balance, his spells are a great asset, especially late in the world of balance on the flying island.Some abilities are still very
Edgar:In the world of ruin he is average at best.In the world of balance, especially early on, he is the most powerful one you will have, aside from sabim, but not so in the world of ruin.One of the things that made him so great, was his sound blaster item.It made everything you fought easy as pie, and auto crossbow dealt a lot of damage.In the world of ruin, its not as strong and sound blaster no longer works.Chainsaw is good damage, but the auto insta death attack that doesn't work on bosses wastes it.It does do a good amount of damage.Drill is good too, as is scan ability.
useful.
7:Cyan:Wether its psycho cyan or some other tricks, he can deal a deadly amount of damage.If he wasn't so slow, he'd be the most overpowered character in the game.15 hits of 9999? he'd be like (knights of the round anyone?) Some abilities are useful, but its just too slow.He needs haste and quick to be useful.He's semi useful in arena, but not that good.
8:Strago:He has a lot of useless abilities, like pep up and explode.Those abilities really don't help me against a boss or enemy.I think some abilities needed to be worked on.Other times:He has great abilities like white wind and grand train and other force field.He is overall still a solid class.
9:Relm:Her painting ability is almost completely useless most of the time.Only time you ever use it is to get lores.She is a good character for spellcasting.
10:Umaro:Mostly useless, except for the tower of fanatics.His rage attack makes him the only one who can attack physically, bypassing the spellcasters easily.Good place to grind as well.

I prefer the following characters:
- Tina and Celes for their magical power
- Edgar for his toys (chainsaw massacre!)
- Mash for his Blitz attacks (powerslam the phantom train!)
- Shadow for his powerful dog

Terra - End of the game she's a powerhouse with offering genji glove + 2 atma weapons (or illumina and atma)
Celes - Same as Terra at end game. Equips powerful swords.
Gau - getting cat scratch (did they gimp it in advance version?) and His speed and general levelability with magic make Him useful throughout the game,
Generally all users can be useful once you get Quick,
Cyan - His Wind Slash sword it can hit for high damage against the whole party. and His damage has always been dependable
Sabin - Though I didn't use bum rush much (it is a good skill) he was good with his end-game equipment
I know the dungeon was split into parts, but my party that fought Kefka where
Terra, for godlike power and speed with swords,
Edgar with autocrossbow
Cyan, for SwordTech
and Sabin. one round and kefka was defeated.

- Terra and Celes (for their magic and strength)
- Edgar (due to his tools ability)
- Sabin (blitzs and double claws)
- Locke (started use his steal ability to get the genji glove, gave him two knives)
- Setzer (started to use his GP toss)
- Strago (mostly gave him black magic)
- Relm (never really used her sketch ability but gave her the fake moustache and white magic)
- Shadow (throwing scrolls and ninja stars)
- Cyan (never really used him much except in the veldt cave and Kefka's tower)
- Mog (only use his charm for the Phoenix cave)
- Gogo (only used him in Kefka's tower)
- Umaro (never used him at all)
- Gau (I don't really like his rage ability, it would take a year to learn them all)

Usefulness is battle is an odd measurement in FF6, because pretty much any character can be trained to deal max damage with any attack in a relatively short amount of time. Without double checking, I think it's the Genji Glove + Offering combo that deals 80,000 damage per turn, and everyone but Gau and Umaro can use it? If we're talking usefulness throughout and not just end game stats, then something like:

1. Edgar :: auto-crossbow pretty much makes the game a joke and you have it out the gate, plus drill is typically the first thing in the game to cap out at 9999 damage
2. Locke :: everything that can't be 1-shot with Edgar can be 1-shot with something Locke can steal
3. Gau :: he's objectively the most powerful character for like half of the game, even if he isn't very fun to play
4. Celes :: big hits + magic, runic slightly more useful than Terra's infrequently-available transformation
5. Terra :: big hits + magic
6. Relm :: sketch can do some wild shit
7. Sabin :: abilities start out powerful and easy to execute. Diminishing returns
8. Mog :: Can dances can be exploited to do stupid damage early? Something tells me the can be and I should have him much higher than this, but I don't remember it.
9. Cyan :: His special ability is basically to make you wait a few seconds for what everyone else can execute instantly.
10. Gogo :: Highly useful in concept, but you don't recruit him until everyone is doing max damage with every attack anyway.
11. Shadow :: Except for this guy.
12. Setzer :: And this guy.
13. Strago :: And most certainly this guy.
14. Umaro :: lol

Pretty much everyone who can Wear genji glove and offering can be turned into a complete game breaker.

1. Edgar - Hits hard, nice defense, well balanced all-around.
2. Celes - Reliable runic is more helpful than morph Terra. A softer version of Edgar. Celes would have beaten Edgar for being faster than him but Edgar's tools always hits harder than her magic.
3. Terra
4. General Leo
5. Sabin - A controllable Umaro. Would be considered the best if he was faster.
6. Umaro - Gets you through the Cult tower, gets you through Coliseum, hits the hardest in a non-grind game. I always get him first as soon as WoR opens up.
7. Gogo - Very fun to use.
8. Locke - Boring character forced on you in WoB. Nice defense and perfect speed though and the whole versatility thing.
9.. Gau
10. Relm
11. Setzer
12. Shadow
13. Cyan - One of my favorites though.
14 Strago
15. Mog - If dance was guaranteed to work he'd be placed at 5.
16. Magitek Armor
17. Ghosts