What are the main katas of each style and why?

[Wado Heretic]

I have to echo the sentiment of Naihanchi. All I would add is that it is important to experiment with it to get the full experience. Look at different versions from what you know, and experiment with trying out the differences and see how they effect the techniques. I like to experiment with Motobu Choki's advice, but also the Shotokan and Wado-Ryu versions of the Kata, which have a different stance and hand movements to the Shorin-Ryu version. I also experiment going back and forth or to the corners with the kata, and with the Bunkai. The key thing, however, is the postural aspect of the Kata and getting that right and carrying those ideas into other kata. The more I study Naihanchi, the more I realise is there, and how little I know about karate and bujutsu.

Aside from that, speaking of other kata traditions, there are a few sources I would recommend looking at:

For combat sports one has to recommend Ashihara, Enshin, and Byakuren Karate, Nippon Kempo, and Shorinji Kempo. The kata in these traditions are done from a fighting stance, and are collections of techniques intended for use in Kumite against an active opponent. Shorinji Kempo is not so involved in the application aspect but still worthy of study for fighting techniques.

Yoseikan Budo, Taido, and the new competition forms the Kukkikwon have created feature dynamic kicks, and techniques not represented in traditional kata, which are worth considering if you want to add such techniques to your repertoire or figure out a way to incorporate them in your shadow-boxing. The Yoseikan and Taido Kata also have examples of kata intended for practicing techniques suitable for combat sports as well, but I do not feel are as accomplished as those I mentioned in that context.

The manner that kata are performed in Egami ha Shotokai-Ryu is also worth looking into if you come from a Shuri-Te background. It is a very different manner of performing kata and a very different mindset I have found benefit in experimenting with, especially when regarding the applications from the Irimi concept. Although I disagree with the global changes, and many of the conceits and assumptions, of Egami: his approach to the Taikyokugata made me recognise their potential value as a training device.

For self-defence techniques, Short Form 3 from Ed Parker's Kempo Karate is a pretty good form to look at. As it is based on self-defence techniques, and built up from those techniques, it can be an interesting study in bridging movements to application.

Simplified Yang Style Tai Chi is something I have practiced, and found a lot of the lessons about posture and structure incredibly useful. Many of the moves also reflect grappling rather than striking techniques, and it has helped refresh and renew my analysis of Karate Kata. A very different take on performing movement to a lot of traditional karate.

[RW]

This article stuck with me:

https://sites.google.com/site/internationalkenshikai/articles-on-karate-essentials/interview-with-sensei-john-rakoto

Among the different schools of Karatedo, Shitoryu has the largest number of kata. Which do you think are the most important, if it is possible to choose?

The fact that Shitoryu Karatedo has a large number of kata, is a great wealth. It is certain that it would take several lives to master all of them. But it is important to note that we have 12 basic kata, which build the core of Shitoryu Karatedo. These are the "Yotsu no kata": dai ichi dosa, dai ni dosa, dai san dosa, dai yon dosa; then the 5 Heian, Naihanchin Shodan, Sanchin and Tensho. Together they form the "Juni no kata," the 12 basic kata characterizing our style. We also have our specificities, kata created by our founder, Kenwa Mabuni Sensei. Each Shitoryu karateka must know them.