Most forms of global skepticism, skepticism about everything, are only hypothetical or methodological. We are not asked to actually withhold belief in everything, we are merely being told that for all we know we may be, e.g., brains in vats being stimulated to think there is an external world. The reason this is just hypothetical is that we are given no reason to think this is actually the case, it's just that the skeptical scenario takes away any reason we could have for thinking it is not the case. Any evidence we could have, any test we could construct to make sure that the world we experience really exists, is just as readily explained by the skeptical theory. (Of course, this is controversial: Hilary Putnam has argued, brilliantly, that on a linguistic-externalist view our words obtain their meaning by virtue of their relation to their object in the world. So our word "vat" means something because there are vats in the external world. But then in order for the brains-in-vats skeptical scenario to be correct, it has to be based on an actually experienced external world, which of course contradicts the scenario.)
In contrast, real skepticism gives you a positive reason for disbelieving, or at least withholding belief in, everything. Plantinga's skepticism is an example of this, but he provides an escape clause: one can always deny naturalism and avoid the skepticism. Perhaps the best example of real skepticism is Agrippa's Trilemma: the question asked is, how is any belief justified? and there are only three ultimate answers we can give. First, we could say that it's justified by another belief, which is justified by another belief, which is justified by another belief... and this chain goes on to infinity. So it's a case of infinite reference. Second, we could say the belief is justified by another belief, which is justified by another belief, which is justified by another belief ... which is justified by the first belief. This is a case of circular reference. Third, you could say the belief is justified by another belief which is justified by another belief ... and that belief requires no justification. This is a case of foundational reference, i.e., it refers to a belief that functions as a foundation.
So what's the problem with infinite reference? Generally, philosophers point to infinite regresses as refutations of positions, but what exactly is the problem with it here? Roughly, the first belief is alleged to derive its justification from another belief, which derives its justification from another, etc. In other words, each step in the chain only has derivative justification. But without some source outside the system to input justification into it, no step will have any. It's like some of the cosmological arguments: imagine you have an infinite number of freight cars connected to each other and ask how they are moving. The first is moving because it's being pulled by the one in front of it, which is being pulled by the one in front of it, etc. But if the chain of cars goes on to infinity then why are the cars moving at all rather than just standing still? It doesn't matter how many cars you add to the chain, without some source of motion, they're not going to move. The only philosopher I know of who defends infinitism is Peter Klein in his "Human Knowledge and the Infinite Regress of Reasons", Philosophical Perspectives 13 (1999): 297-325, "When Infinite Regresses Are Not Vicious", Philosophy and Phenomenological Research 66 (2003): 718-29, and elsewhere.
What's the problem with circular reference? Essentially, it's the same problem: each step in the process is justified in virtue of its relation to the previous step, that is, it has derivative justification. Circling back to a step already referred to in the process does not somehow bring the needed justification into the picture. As Victor Reppert writes, "Circularity is the epistemic equivalent of counterfeiting" ("Eliminative Materialism, Cognitive Suicide, and Begging the Question", Metaphilosophy 23 : 386), since it gives the illusion of providing a source of justification without doing so. Nevertheless, circular reference is vastly more popular a position than infinite reference, in the form of coherentism. As far as I can tell it's still the minority view, but it's defended by many of the top philosophers around: Keith Lehrer, Nicholas Rescher, the early Laurence BonJour, Brand Blanshard, names could be multiplied. These are some of the smartest people of the last century, so coherentism cannot be dismissed without interacting with their writings.
(Another interesting claim that I've read about but have not read any actual proponents of is that a belief does not only derive justification from another belief but from the actual derivation process. So even if the beliefs themselves have no original justification, if you have enough steps involved, you will eventually build up enough justification that the beliefs will become justified. This could potentially rescue both infinite and circular reference.)
What's the problem with foundational reference? Historically, foundationalism has been the near-universal position among epistemologists, and as far as I can tell, is still the majority view today. Some beliefs simply don't need justification, or they carry their justification in themselves, they are self-justifying. The problem here is dogmatism. To say that some beliefs are the ground level, to say that some beliefs don't need to be justified by something else is to say that we don't need to question them, we don't need to verify them. But virtually every class of belief that has been proposed for this position has been challenged precisely because they can be. We need to have a reason for a belief and to continue believing in the absence of a reason is mere dogmatism. Again, most epistemologists would disagree with this, they would say that there are some foundational beliefs and that not having a reason for a belief does not make it irrational or unacceptable in this case. Some, such as Plantinga, seek to escape the charge of dogmatism by making these beliefs defeatable: they can be questioned, they can be challenged, they are just innocent until proven guilty.
So Agrippa's Trilemma says there are three options -- infinitism, coherentism, and foundationalism -- and none of these are acceptable. Therefore, we have a reason to reject each one, and therefore to reject the possibility of having any knowledge whatsoever. Thus, we are left with actual skepticism, not hypothetical or methdological skepticism. The only alternative is to do what virtually all epistemologists have done: accept one of three options.
(cross-posted at Quodlibeta)