Take a look at http://www.youtube.com/watch?v=bEX7BnD595A. To write about this topic would take several hundred - at least - words, and would result in so many opinions.
It is all a matter of persona preference. Of course, for me the case is closed badger is best! Let the posts begin!!!
Any of the three can become for any one individual a favorite My lather monster is a 3 dollar synthetic with nylon bristles it whips up anything massivly, I've got badgers I use mainly with creams and boar with cake soaps.
there are other parameters to observe loft, shape, knot width stiffness, even hybrids like badger/boar combinations. it's not as simple as synthetic vs badger vs boar
While some of it does depend on individual preference, there is something to be said for one purely objective characteristic that has a huge effect on the production of lather -- water absorbtion and retention. Badger just picks up from the sink (and transfers to the lather bowl) a lot more water than boar. I don't know why that is, but it's something that must be related to the particular characteristics of the animal hairs, because it's true for every grade of badger and boar. Maybe it makes badgers better swimmers than boars, who knows. But what I've noticed is that, when you are using a badger brush and getting thin, runny, soupy lather, you can cut down the amount of water in your mixture automatically by just switching to a boar brush. Because boar really doesn't pick up very much water at all, it's actually pretty easy to whip up great lather with a boar brush in most situations -- and you can always add a little bit of water if you need it with your fingertips (whereas removing an excess of water is pretty much impossible).