After attending an Oberwolfach workshop with Kristin DeVleming and Anne-Sophie Kaloghiros where they helped me navigate the literature on K-stability for Fano 3-folds (thanks so much!), the current state of the art can now be found on the K-stability page.

I will add more explanations to Fanography after my holidays, for the time being I'll just refer to the K-stability Wikipedia page.

I am sure I have made mistakes, and I'm sure I've missed recent developments. Please get in touch if there is something I can improve!