...besides "nothing" for you English purists, that's it.
Since
that discussion is raging, I thought it'd be a good time to talk about what the Touhous are actually doing that justifies the kind of convoluted text ZUN uses.
I think that what's happening there is a mix of innate skills with most of them also studying "magic". Scary quotes because "magic" can be a lot of different formal disciplines ("Shrine Maiden training" for Reimu and Sanae, Martial arts for Youmu, plain old Magic for Marisa and Alice, etc) that have the same end-effect: gives breadth of power to the character.
So, for example, if they need to "slow down time", Sakuya or Kaguya can do that directly with their innate abilities. Yukari and Sanae can use their vague innate abilities and do the same. But anyone with any formal training at all can hack some kind of solution. Yes, really, even Youmu or Meiling. Also yes, this means that someone like Yukari has several ways to do the same thing.
Add to this mix the effect of some powerful races. "Being a Yama" or "Being an Oni", is pretty much the same of having a hugely useful innate skill. Finally, any being who gets very old (like 1000+ years) ends having the same breadth of power than a magic user. I don't know if this is because they found a time in their milenar schedules to study or if "being around for a while" already counts as a formal discipline. Whatever. I did the conservative thing and noted them as both being Elder <thing> and knowing magic.
Here's my interpretation to what the characters can actually do, by game appearance order. I'll divide what's a innate ability and what's formal training. I won't grade how well they are trained to avoid turning this into a LOL POWERLEVELS discussion. Take your own conclusions.
Reimu: Innate ability to fly (away from reality); Innate ability to eat sweets and don't get fat; Formal training as a shrine maiden.
Marisa: No innate ability; Formal training with magic.
Rumia: Innate ability to control darkness; No formal training.
Daiyousei: Innate ability to remain unnamed.Cirno: Innate ability to control cold; No formal training.
Meiling:
Innate ability to turn into a dragon;No innate ability; Formal training with martial arts/qi gong.
Patchouli: No innate ability; Formal training with magic.
Sakuya: Innate ability to control time; Formal training with magic/some kind of knife throwing/assassination martial art/whatever. (Sakuya, more than most Touhous suffers from "too many backgrounds!")
Remilia: Innate ability to control fate; Vampire; Formal training with magic(?)
Flandre: Innate ability to destroy anything; Vampire; Formal training with magic (500 years old magical girl, lol)
Letty: Innate ability to control cold; No formal training? (really, Letty is down there with the Akis in the "characters people don't talk about" category)
Chen: No innate ability; Formal training with magic.
Alice: No innate ability; Formal training with magic; Formal training with doll manipulation
The Prismrivers: Innate ability to play mind affecting music; Formal training as musicians
Youmu: No innate ability; Formal training with martial arts.
Yuyuko: Innate ability to manipulate death; Elder Ghost; Formal training with magic.
Ran: No innate ability; Elder Youkai; Formal training with magic.
Yukari: Innate ability to manipulate boundaries; Elder Youkai; Formal training with magic.
Suika: Innate ability to control density; Oni; Formal training with magic.
Wriggle: Innate ability to control insects; No formal training.
Mystia: Innate ability to confuse/blind people with song; No formal training.
Keine: Innate ability to control history; Formal training with magic.
Tewi: Innate ability to grant luck; Elder Youkai; Formal training with magic?
Reisen: Innate ability to cause insanity; Innate ability to control waves?; Lunarian; Formal training with lunarian knowledge.
Eirin: No innate ability; Lunarian; Formal training with lunarian knowledge.
Kaguya: Innate ability to control the eternity and the instantaneous; Hourai Immortal; Lunarian; Formal training with lunarian knowledge.
Mokou: No innate ability; Hourai Immortal; Formal training with whatever she did to get that badass. (my personal theory is that she melded with a Phoenix)
Yuuka: Innate ability to control flowers; Elder Youkai; Formal training with magic.
Medicine: Innate ability to create poisons; No formal training.
Komachi: No innate ability; Shikigami; Formal training with celestial knowledge.
Shikieiki: No innate ability; Yama; Formal training with celestial knowledge.
Shizuha: Innate ability to control autumn leaves; Kami; No formal training.
Minoriko: Innate ability to control harvests; Kami; No formal training.
Hina: Innate ability to control misfortune; Kami; No formal training.
Nitori: No innate ability; Kappa; Formal training with engineering.
Momiji: No innate ability; Tengu; Formal training with martial arts.
Aya: Innate ability to control the winds; Elder Tengu; Formal training with magic.
Sanae: Innate ability to cause miracles; Formal training as a shrine maiden.
Kanako: Innate ability to create sky; Elder Kami; Formal training with magic.
Suwako: Innate ability to create earth; Elder Kami (maybe not a Kami at all, but something older); Formal training with magic.
Iku: No innate ability; Formal training with celestial knowledge.
Tenshi: No innate ability; Celestial; Wielder of the sword of Hisou; Formal training with celestial knowledge.
Kisume: Innate ability to control will-o-wisps; No formal training.
Yamame: Innate ability to control diseases; No formal training.
Parsee: Innate ability to control jealousy; No formal training(?)
Yuugi: Innate ability to control supernatural phenomena(
); Oni; Formal training with magic.
Satori: Innate ability to read minds; Formal training with magic.
Orin: Innate ability to carry corpses(?); No formal training(?)
Okuu: Innate ability to control nuclear force; No formal training(?)
Koishi: Innate ability to manipulate the unconscious; Formal training with magic.
Nazrin: Innate ability to locate items; Elder Youkai; Formal training with magic(?)
Kogasa: Innate ability to surprise people; No formal training.
Ichirin/Unzan: No innate ability; Elder Youkais; Formal training with Buddhism.
Murasa: Innate ability to sink ships; Elder Youkai; Formal training with Buddhism(?)
Shou: Innate ability to gather treasures; Elder Youkai/Avatar of Bishamonten; Formal training with Buddhism.
Byakuren: No innate ability; Elder Youkai; Formal training with magic and Buddhism.
Nue: Innate ability to make forms unknown; Elder Youkai; Formal training with magic.
Hatate: Innate ability to use spirit photography; Formal training with magic.
Kyouko: Innate ability of reflecting sounds; No formal training (will learn Buddhism).
Yoshika: Innate ability of eating anything; No formal training.
Seiga: Innate ability of passing through walls; Elder Hermit; Formal training with Taoism.
Tojiko: Innate ability to control lightning; Elder Ghost; Formal training with Taoism.
Futo: No innate ability; Elder Hermit; Formal training with Taoism.
Miko: Innate ability to listen to the people's desires; Elder Taoist Saint; Formal training with Taoism
Mamizou: No innate skills; Elder Tanuki; Formal training with magic.
When compiling this list, something became clear: Sometimes ZUN cheats/trolls and writes the "ability" info as something that comes from the character race or training. Sometimes, as with Youmu, or Ichirin/Unzan, it's a bit worse, as the ability text is literally: "ability to do exactly what you're seeing on screen!" Thus, people like Shikieiki, Komachi, Alice, Mamizou or Futo get "no innate skills". Mamizou is a shapechanger
because this is what a Tanuki is, while Alice or Futo are just being described as doing their jobs. Finally, Tenshi seems to be the only case in Touhou (at least in the Windows era, I don't know about PC-98) of a character deriving most of her skills from an item.
Regarding the Kappa and Tengu, I gave "no innate ability" to Nitori and Momiji because I assume all Kappa can control water and all wolf Tengu see very far. But I gave "control of winds" to Aya because Hatate is also a crow Tengu and doesn't have that.
Finally, Symposium of Post-Mysticism is about to be released and I HOPE it fills the holes I have in my knowledge about the later games (seriously, I have no idea of what kind of people the girls of the Palace of the Earth Spirits are).