Hi.
Yes, there are definitely sets of specific kata for each ryu... that's the primary method of transmission and practice. While there are a number of groups of each ryu you mention, with some variation to their syllabus (depending on the teacher and their lineage), there are more similarities than differences in the main.
We'll start with Muso Shinden Ryu, as that's the ryu I have more experience with.
Muso Shinden Ryu is, pretty simply, a modern reworking/line of Eishin Ryu Iai, along with Muso Jikiden Eishin Ryu (realistically, they are two different lines of the same school... and share the majority of their syllabus). The school is broken up into three primary sections, with some lines having additional sections, such as paired practice, or even jujutsu methods.
The main three sections are:
- Shoden Gata/Omori Ryu. This section is made up of 12 kata (Muso Jikiden Eishin Ryu only has 11... with a variation sometimes taught, which is a formal kata for MSR). These techniques almost exclusively begin from seiza, a formal kneeling position sitting on your heels (one kata, Koranto, is a standing one). The names are different from the ones used in MJER.
- Chuden Gata/Hasegawa Eishin Ryu. This section has 10 kata, with most being performed from a seated position called Tate-hiza. This is the "original" form of the school, with the Omori Ryu being a later addition.
- Oku Iai. The final standard set is the Oku Iai (Inner Iai), and consists of both Tate-hiza no Bu and Tachiai no Bu, or Suwari-waza and Tachi-waza. The Suwari-waza (seated techniques) consists of 9-11 techniques (depending on the line and how you count them), and the Tachi-waza (standing techniques) consist of 11 or 12 kata, again depending on the line.
Additional sections include Tachi-Uchi no Kurai and Tsume-ai no Kurai (two sets of paired forms done as a style of kenjutsu), and, very rarely, areas such as Tosa Iai Jujutsu, including Daisho Tsume, Daisho Tachi Tsume, and Tsume no Kurai.
It's not common to find much beyond the three main sections, but historically, the other aspects were part of the school's teachings.
Mugai Ryu is a bit more complex, with a much wider array of lines having been created over the years... the biggest "main" group these days is the Meishi-ha... although there are a number of other groups around. I haven't had much experience with the Meishi-ha Mugai Ryu, but a quick search indicates that the syllabus is primarily as follows:
- 20 Iai waza, divided up into sections such as Go-yo, Go-ka, Go-o, and Hashiri-kakari, covering both Seiza no Bu and Tachi no Bu.
- Naiden waza, a set of advanced techniques taught to Menkyo practitioners. These focus on controlling the opponent, rather than lethal force.
- A range of kenjutsu, covered in both Katana no Kata and Wakizashi no Kata.
There are members of Mugai Ryu on the forum, so hopefully they can clarify or add to my comments here.
@pgsmith ?