Welcome to my Special Skills guide, in this guide you will find tips and recommendations about what skill can be learned by which of available characters.
If skills are tied together (like Sawblade and Triple Sawblades or Spark and Big Spark) they will be under the same section.
Under each skill there is list of characters that can learn it and the one recommended has it's name bold. Example - Anna/Matt/Lance.

Offensive skills may be a lot easier to decide who should learn what. So let us begin:

Hellfire:
Anna/Matt/Natalie
This skill can be effectively used by either Anna or Natalie. Anna doesn't have any fire spells by default and this is one of the two available to her, so even though she is focused a bit more on physical damage than magical it can help her cover more elements. On the other hand, Natalie's Fireball spell has only 120 power while Hellfire has 220 on last level which is nearly twice as strong, so by teaching it to Natalie it will greatly increase her single-target damage output to nearly every enemy that doesn't resist fire.
Note - Keep in mind that Anna must learn Berserk first which is also a special skill.

Fire Spin:
Anna/Matt/Natalie/Lance
This is the second fire-elemental skill available to Anna, and while the previous one can be learnt by Natalie with great effect, there is no reason to learn this one to anybody but Anna. Matt is focused on physical damage, putting magical spell with just 50 power on him is just pointless, both Natalie and Lance have their own multi-target spells with greater power. Anna is the best user of this spell as its defence-debuffing effect is great for setting up a combo with Matt and Anna's physical skills.

Spark and Big Spark:
Anna/Matt/Lance
This one is an insta-learn for Matt as he doesn't have any other thunder skills, and this one is additionally physical. Anna already has her Spark Arrow with nearly the same power and Lance, who is focused a bit more on magical damage than physical, has his Plasma Field.
Note - this skill is 80% elemental while Matt's other skills are only 50%, so it will have much better effect when used against enemies weak to it than Matt's other skills.

Plasma Cage:
Anna/Matt/Natalie/Lance

Icicles:
Anna/Matt/Natalie/Lance
I found this spell pretty much as "last resort" as most of the time there is no reason to use it. Matt has his Iceberg which is stronger, Natalie has nearly no physical attack so it is pointless, Anna has her Frost Arrow which may be a bit weaker, but is more consistent, has higher element - 80% while Icicles has only 50%, and inflicts a better status effect. Only left is Lance who has only one other ice attack so even though it is physical it will help him cover more elements.

Blizzard:
Anna/Matt/Natalie/Lance

Hailstorm:
Anna/Natalie/Lance
Lance is recommended as Natalie already has her own multi-target ice skill, while Anna favours physical attacks instead.
Remember that this skill damages your party as well, although damage is very minor. If you have over 100% ice elemental resistance you can use this attack to heal your party slightly, or alternatively, you can damage your team to charge their limit breaks faster.

Gaia Seed, Gaia Bloom, and Gaia Blossom:
Anna/Natalie
This skills are magical, and earth elemental, Anna doesn't need that, she already has her physical earth skill and she has more attack than magical attack, instantly after finding Natalie just make Anna forget this skill and teach it to Natalie as this are the the only earth spells available to her.

Sand Dune and Giga Drill:
Anna/Matt/Lance and Matt/Lance
I put those skills together as it will be easier to explain this way. Both of them are the only earth skills available to Lance, but both of them are physical. Because Giga Drill has only 25% element and its defence piercing effect scales very good with attack I recommend Matt to learn it. Sand Dune is left for Lance so he will have at least one earth elemental skill, there is no reason to put it on Matt or Anna anyway as their Cataclysm and Lumber are much stronger.

Toxic:
Anna/Natalie
It is magical and has only 50 power while Anna's Vines have 180, pointless on her. While it is the only poison skill available to Natalie so just teach it to her, it may be weak but has increased chance to poison the target which can still prove useful against most bosses that don't resist poison.

Acid:
Anna/Matt/Natalie/Lance

Bamboo Trap:
Anna/Matt
Insta-learn for Matt, it has exactly same power as Anna's Vines while being 3 times stronger than Matt's Nettle, additionally its increased accuracy may be useful as Matt's accuracy is slightly below average.

Blast and Big Blast:
Natalie/Lance
Lance already has his own bomb-elemental skills while none of the other characters does, it is pointless to teach it to him. By giving it to Natalie you will have two characters with bomb skills instead of one.

Airwave and Shockwave:
Matt/Natalie/Lance
Natalie and Lance will make far better use of those skills than Matt, as they deal magical damage. It is the only wind-elemental skill available for Natalie, and because Lance already has access to Power Metal then it is recommended for Natalie to learn it.

Power Metal and Death Metal:
Matt/Lance
As a magical skill, it naturally will be more potent in Lance hands. Although Power Metal is weaker than Shockwave, it heals the party and damages enemy at the same time, which makes it more useful.

Bubbles and Bubble Blast:
Natalie/Lance
This one is purely a matter of preference, as neither character has other water skills available. You may want to teach it to your primary damage dealer to make full use of its damage, or the other way around - use it with the weaker character to wet the targets and thus increase the damage of stronger character's thunder and ice-elemental skills.

Geyser and Flood:
Anna/Matt
Matt will start the game with Geyser already learnt. Since it is the only water-elemental skill available for Matt, and Anna already has her Aqua Arrow (which is stronger than Flood anyway), then simply leave it as it is and enjoy the rainbow.

Talisman Seal:
Anna/Matt/Natalie
This skill has relatively low power and all three characters already have their own holy-elemental skills that are far stronger. I recommend Anna to learn it, that is because her Soul Arrow always copies targets buffs (and debuffs) and replaces your own, so you will often encounter situations where it is simply not worth using, that is when Talisman Seal comes as an useful alternative.
Talisman Seal's great strength lies it the fact, that it nearly always inflicts its special effects (which is Syphon and magic attack debuff), which comes very useful when facing foes that rely on magical attacks.

Holy Fire:
Anna/Matt/Natalie/Lance
As mentioned above, Anna, Matt, and Natalie already have their own powerful holy attacks, Lance however doesn't have any holy-elemental skill available except for this one. Quite simple choice, right?

Dark Flare:
Matt/Natalie/Lance

Fright and Screamer:
Anna/Matt

Sacrifice:
Anna/Matt/Natalie

Absolute Zero:
Anna/Natalie

Tsunami:
Anna/Matt/Natalie

Ion Cannon:
Matt/Lance
It deals magical damage so it will be more potent when used by Lance rather than Matt.

Supernova:
Natalie/Lance
Lance already has his Nuke, which is quite similar (and even deals higher elemental-based damage) so Supernova will probably end up being taught to Natalie.

Black Hole:
Natalie/Lance

Annihilate:
Anna/Matt

Power Blast:
Anna/Matt
Matt already has his Cleaver, and giving Anna a limit break that uses her weapon's element and special effects is probably a good idea.

Rapture:
Anna/Matt/Natalie/Lance
Rapture deals magical damage, so it naturally will be more potent when used by Natalie and Lance. Matt and Anna already have their weapon-dependant limit break anyway (Cleaver and mentioned above Power Blast). Whether Natalie or Lance should learn this skill is a personal preference, it is most likely a good idea to teach it to the one that is your primary damage dealer.

Work in progress, missing descriptions coming soon.