Hmm...rough thoughts on the best class at using each skillset. As a general rule, this does NOT consider the physical damage the class does. (Sometimes it comes up anyway, like Charge).
Basic Skill:
Squire/Ninja/Knight/Bard
I'm assuming it's being used mostly for Yell/Heal, which means you only really care about speed, mobility, and durability.
Item:
Chemist/Ninja/Knight/Bard
Chemist has innate Throw Item. Past that, speed/durability are your main concerns.
Battle Skill:
Chemist/Thief, Mediator/Ninja)
You're looking for range or break twice. Thief gives you more HP than Ninja (and PA doesn't matter for breaks), Chemist gives you essentials that Mediator fails at (revival, etc). That said, Ninja/Mediator are both faster, so arguably better.
Charge:
Bard/Mediator/Chemist/Ninja:
Charge more exploits certain aspects of a class than gets used by that class' stats. In Bard's case, Harps have crazy WP now. In Mediator's case, we just added innate: Train, which combines extremely well with Charge. Chemist...guns with a skillset. Ninja: highest damage in the game, in case you're not killing assassination missions.
White Magic:
Wizard/Geomancer, Squire/Priest, Knight
For max damage/healing, best practical speed with MP-based revival, and unkillable healer respectively.
Black Magic:
Wizard (Summoner, Geomancer, Squire/Priest, Knight)
Fun trick about 18 CT death: it doesn't take two full turns if you have 5 speed...which Summoners do at low levels. Though...yeah, mostly you want to just Bolt stuff dead, which means Wizard (and I guess arguably Geo at endgame).
Punch Art:
Dancer, Ninja, Monk
Two classes can have Martial Arts + Attack Up. Monk, naturally, and Dancer. I recently talked Laggy into adding some Dancer-unique PA boosting (cloths)--now female Dancers they can compete with male Monks. (Typically more damage with Earth Slash, but less move/jump/speed/HP. Worse everything at level 99, though). Ninja doesn't have innate Attack Up or innate Martial Arts; Ninja just smokes both classes for stats.
Steal:
Knight, Ninja, Bard
Percentages be damned--faster = more chances to steal. (Monk/Dancer technically get the highest percentages). Knight's also durable--not being dead helps.
Yin Yang Magic:
Squire, Knight, (Geomancer, Priest)
YYM cares about three things: Speed, Move (especially now that Life Drain is range 1), and non-crap MP. Ninja/Bard does speed, but fails the non-crap MP. Squire is tied for #2 speed, #1 move, and fairly high up on MP (Priests beat). Knight's just that durable, and has enough MP. Geo's got plenty of everything but speed, but being a charge skillset you may want to make yourself slower.
Time Magic:
Wizard, (Priest, Squire, Summoner)
Okay, everyone uses Haste pretty good, so I'm focusing on the serious resource intensive Meteor/Quick stuff. Wizard's always at least tied for the best MA (well...Time Mages would have beat nerfed Wizards briefly in Chapter 2 if Poison Rods hadn't been given a purpose *shakes fist*). Speed's the other thing I look for when, say, trying to Meteor the Ninjas at Yardow (Speed 8 needed). Priest does that better (Wizard Staff gives them +1 MA over Squire, and they have more MP). On paper Squire does this job better in Chapter 4 (Rune Blade one-ups Wizard Staff, MP equipment is easier to come by, and the MP gap is 110 to 120). Summoner...they do have the most MP, and sometimes you need to twink speed downward.
Elemental:
Geomancer
This used to be almost close. Almost.
Jump:
Geomancer, Ninja, Squire, Knight, Lancer, Bard, Archer
Jumpers want to be fast...but not too fast (faster than the enemies = fail, Ninja/Bard). Lancers like weapons with the most WP...which would be Harps (sadly Bard PA fails so much that this is questionable), followed by Knight Swords (good, but rare) and Axes (Squire and Geo have great stats for this). Oh, obviously Spears; though you can learn Equip Spear. (Randomly, except for Ninja, the best classes to stick Equip Spear on already had Axes/Knight Swords as an option). Oh, tossed Archer in there in case you learn Level Jump 8, but haven't learned any vertical abilities (Ignore Height innate).
Talk Skill:
Mediator, Ninja, Bard
Mediator has innate Monster Talk, Ninja/Bard has speed.
Summon:
Wizard, Geomancer, (Priest, Squire, Knight)
Summon generally wants the most MA, and a good bit of MP. I've already covered the short-charge meteor angle, which covers the big summons (and why you might consider Priest/Squire to make the timing work). Knight...Golem produces a shield that's equal to your HP total.
Dance:
Knight, (Dancer, Ninja)
Once you start dancing, generally all that matters is durability. Best Wiznaibus damage...with the carpet change now comes from Dancer rather than Ninja. Ninja does still have speed (start Nameless Dance sooner).
Sing:
Knight, (Wizard, Bard)
Again, durability is the big thing you notice. Wizard will do the most healing/MP healing. Bard...starts the song sooner.
Throw:
Ninja, Squire, Bard
Throw is based on two things: Speed (damage), and Movement (range). Bard > Ninja > Squire for speed. Squire > Ninja > Bard for movement.
Side note: hadn't realized it, but yes, assuming 100 growth, Bard's ahead of Ninja for speed from levels 1-12, 18-26, 36-41, and a couple others. Ninja wins at high levels...or if it's a female Ninja with either H-Bag or Setiemson (H-Bag is all that's needed)...or during practical gameplay because Ninja has better speed growth. And Ninja has more damage. And Ninja has a good non-speed stat. Yeah...seems okay to me. Still...O_O Harps.
Draw Out
Geomancer, Squire, Wizard, Knight
Shortly after you finish getting Samurai JP, Rune Blades probably show up (which makes Geo a lot more attractive than Wizard; Wizard's still worth mentioning in case you get done early). Draw Out also cares a lot about move, being a short-range skillset. Geos do move. So do Squires, and Squire extra speed is also welcome. Knight...invincible healer, and Kiyomori doesn't depend on offence stats, and DO involves getting up close and personal.
Math Skill
Geomancer, Wizard, Squire, Knight, Bard, Ninja
I was all excited that Time Mage would get a moment in the sun, what with a secret staff that strengthens holy--but then I remembered MS doesn't have Holy now, D'oh. With those unlocks, safe to assume MS = Chapter 4 = Rune Blades, so Squire's just better than Priest as speed-class. Excalibur's still plenty tempting on Knight without the Holy stuff mattering. Bard and Ninja are fast. You probably want Flame Shields all around now, but Equip Shield exists.
EDIT:
Scorecard--I gave 5 points to whichever class was first in the list for a skillset, and 1 point to the rest. Note that this only measures stats, and for the most part ignores the class physical attack (except Charge and Battle Skill). While I did do calculations, some of this is gut reactions which might be off. Not every skillset is of equal importance. And...some classes with good stats don't show up here at all because they're not #1 in anything (Time Mage is almost as good as Wizard...yet not even on this list).
Knight: 24
Geomancer: 24
Wizard: 23
Squire: 18
Ninja: 15
Bard: 13
Chemist: 11
Mediator: 7
Dancer: 6
Priest: 5
Summoner: 2
Thief: 1
Monk: 1
Lancer: 1
Archer: 1
Might seem like this is centered around a small number of classes, but...honestly? Standard FFT would probably see 35 points for Wizard, 40 points for Ninja, 15 points for Chemist, and...everyone else in single digits if that.