Demnstrating the premises of an argument is more important than demonstrating their link to the conclusion
Here's a major criticism that I think can be applied to how most arguments are made. If we say the structure of the argument is like
if A, B and C then Xmy criticism is that people primarily try to justify the argument by justifying the then part.
They may make some efforts to justify the if part - to justify A, B and C, but I think mostly this is more about giving the appearance of having checked that task off than a genuine attempt to ensure they're true.
Having taken A, B and C to be the case, they argue why X should follow, and try and back that up other reasons why we should take X to be true.
The problem with this is, I think, is that A, B and C actually being the case is far more important to X being true than the line of reasoning that allows you to go from A, B and C to X.
A, B and C define a terms of reference, a picture of the world. The arguments for then X are really along the lines of "in a world where it is such that you have A, B and C, it would follow that X". And I don't think it's that hard to have distorting simplifications in your terms of reference that allow you to derive X, even if X isn't actually true.
Justifying the if part is the major task in showing that the world actually has the certain properties, and works in the particular way that, when you work out the consequences, you see that X is the case.
I think we usually think of the premises of an argument as simply facts, but really the bulk of the premise is an outlook, a framing, a kind of model of the world and the kinds of ways you can reason from facts to conclusions. Kuhn's insight about paradigms is about this kind of thing.
This is a major reason why in most arguments the parties end up talking past each other. Of course the other person's view seems wrong - they're working from different sorts of premises. If you really want to get somewhere, examine the premises.
I can't think of a forum (the popular media, books, academic papers, discussion forums, etc) where it is really considered acceptable to really look into premises, or even to look into the premises behind other people's work. Philosophy is an area where you might expect to see it, though from the stuff that I've seen it doesn't seem to happen.
I suppose blogs are one medium where, in the first place, you can get it out there, and get away with it unscathed, though I kinda doubt anyone would be that interested in reading it.