The Clan’s School of Swordfighting is rarely called by name. In general there are no truly distinct schools, and more one school with a diverse array of styles. This comes from the Clans tendencies to adapt any style of Kenjutsu and the like they considered useful and conserve it, evolving their own very flexible styles of the sword by learning from everyone else.
As one said it, the Schools of Modern Day are exactly that, Schools. They teach about their style. The Clan teach about how to fight.

Generally the Clan’s Swordfighting Styles are summarized under the "Mumei no ken" name, The Nameless Sword.
Kat (and Yumi’s) brand of Iaijutsu is thus considered a specific style within the Mumei no ken School, the Hagane kaminari-ryu (鋼雷流), the Steel Thunder Style. This is is said because the Hagane kaminari style is said to be the fastest of the Mumei no ken Styles, so fast that unless you pay close attention you may miss the lightning until you hear the thunder coming after it.

Another example would be Sazae’s Fukutsu no ishi-ryu (不屈の意志流), Indomitable Will, style. This is a more traditional and conventional battlefield style, meant to be used by those Warriors who will see most combat in a battle and refers to the fact that is is a highly effective style, geared to not allow overexertion while doing precise attacks at the opponents, making its users the most enduring of the clan, allowing them to stay standing while their opponents break down from exhaustion.
A third form is Ippiki ōkami-ryu (一匹狼), the Lone Wolf Style. Its is a variant of Fukutsu no ishi-ryu focused on single combat, designed to use Fukutsu’s battlefield-duel style for a single lone warrior. It is more reckless, focused on a possible quick win. Essentially, if Fukutsu is meant to allow a Warrior to fight a long time, Ippiki is meant to be used then the Warrior is already exhausted from travel and on his own.

To put it in „modern“ terms, Hagane Kaminari-ryu is a „Damage Dealer“ style, Fukutsu no Ishi-ryu is a „Offensive tank“ style.

Other notes:

最初の雷 - Saisho no kaminari. The First Lightning, a opening strike of the Hagane kaminari-ryu. The first lightning is never seen until you hear its thunder.

最後の雷 - Saigo no kaminari. The Final Lighting. A finishing strike of the style

分割雷 - Bunkatsu kaminari. The Forked Lighting. An attack sequence.

百回の落雷 - Hyaku kai no rakurai. The Lightning Strikes a Hundredfold. The Master Technique of the Style.
Drawing the sword and swinging it in one long strike forming a series of slashes. It is ONE strike, not a series of them. This is done by moving the sword in circular motions, forcing the wielder to use extensive footwork and body movement to accommodate this technique.
The crux of the technique is speed, due to the need body motion and that it looses effectivity if the technique takes too long, as it should never take more than a breath to perform. You need to be fast to make it work without giving openings the target can exploit or react to the slashes at speed. The better you are the more slashes you can bring in the same time.

Not all techniques are named, only a few key strikes which are meant to be the basis of variants to be used.



It should be noted that most clan Swords have a solid foundation in all styles, while Spears (the team players) use both Fukutsu no ishi-ryu and Hyaku no yari-ryu (百の槍) (The Hundred Spears) style. The latter is also a variant of Fukutsu, emphasizing teamwork the strongest of the Mumei no ken styles.
so while Sazae uses Fukutsu as primary, or core style, she is fully capable of using both Hagane and Ippiki if the situation requires it. The older a clan member the more versed they usually are in the various styles. So while Kat *now* primarily uses Hagane style, in time she will also develop other styles to add to her arsenal.