Tricky one this, because it depends on who you ask as to which learning style you go for.
Howard Gardner came up with one of the earliest theories - He originally went for about 7 different types of intelligences, but later added at least two more. These included things like physical, interpersonal, intrapersonal etc...
Honey and Mumford was also in fashion for a while. They argued that we had different learning styles and grouped people into four different types - including reflector etc...
The VAK method you refer to is probably the most popular version currently (they use it in my college), That was devised by Dunn and Dunn. So you'd have to look up them for more info.
According to Wikipedia, there are over 80 different models - I don't know if that's true but when I did a module for my MA on teaching and learning, we looked at a lot of different versions!
Trouble is that there is very little evidence for any of them. One of my lecturers who specialised in applying neuroscience to education argued that there was virtually no evidence for learning styles whatsoever, and when I did an essay on this, I found there to be very little substantial evidence too. This is explained in this article from wikipedia (not the most relaible I know, but it is in line with what I read during when I studied this issue) en.wikipedia.org/wiki/Learning_styles
It is very popular in schools at the mo, but not sure there is an awful lot of science behind it. I know many of the psychologists at college totally dismiss it. And of course, there is the problem that even if we do have different learning styles, which model do we follow? Can we really pigeonhole people into 3 categories so neatly? Or is it the 7 different multiple intelligences? Or is it four different learning types???
HTH