Before I can have an opinion on whether science can address claims about the supernatural, I have to have a definition of supernatural. Unfortunately, it is not really clear to me what it is supposed to be. People seem to have very different ideas of what they mean with it. Or perhaps the definition is mutable, to be changed however it currently suits those who want to argue that science should keep its muddy paws off their cherished beliefs? Let's try to nail it down by looking at possible definitions on offer, see what sense each of them makes, and whether the definition really implies an inability of science to address related claims.
At the beginning of the Wikipedia article on the subject (accessed 20 Feb 2013), two definitions are presented. The first is
(1) that which is not subject to the laws of nature
Hm. So that assumes that there are things in existence, presumably things that we can see and know about, and those things are subject to the laws of nature that are clearly what science is about. And then there are other things that exist but are not subject to the laws of nature, and presumably the idea is that because science deals with describing (only) laws of nature it cannot deal with those other things.
Now I would first ask if we can we observe that which is not subject to the laws of nature. I assume yes, because if not then nobody knows anything about them, be they scientist or not, and a lot of people do claim such knowledge. Note for example that miracles and ghosts are generally assumed to be observable in some way, and that souls and gods are generally assumed to have an influence on the world. In other words, we can observe those phenomena.
The second question is then whether it is assumed that the supernatural does not follow the laws of nature because it follows its own laws or because it does not follow any laws whatsoever. If it is the former, it is unclear to me in what sense science could not try to deduce the laws and regularities. The supernatural would be observable, it would follow some discernible logic, and thus the scientist can get to work and at least fare no worse than with black holes or superstrings.
If it is the latter, then we run into the problem that nobody can understand the supernatural in principle. But really that is clearly not what people mean when they talk about the supernatural anyway; they don't believe in gods, spirits or souls because they assume that they will behave in a completely unforeseeable, capricious fashion. Instead believers claim that gods want this and reward that, that souls carry this or that personality trait or memories, etc. The worst one could say is that supernatural beings, such as gods or demons, might be hard to understand because they are postulated to have personality and intelligence, but in what way would science then have more problems dealing with them than psychology, sociology, ethnology and history have with dealing with humans?
So far I am not convinced. The second definition of supernatural suggested by Wikipedia is
(2) that which is said to exist above and beyond nature
I understand even less what that is supposed to mean. Something exists but it is not part of nature. Naively, I can think of only two reasonable definitions of nature: All that actually exists (as opposed to fantasy), and all that exists but has not been produced or thoroughly transformed by humans (as opposed to culture, artifacts and anthropogenic landscapes). Under the first definition of nature, the supernatural does not exist by definition because nature is by definition all there is. I am quite happy with that because as mentioned I do not see the point of the entire concept. Under the second definition of nature, the supernatural would be a synonym for human-made, and that is obviously nonsense because clearly not what people talking about the supernatural have in mind.
There is, of course, a third option, and that is to define nature as that which exists but is not supernature. Fine, but then we are moving in circles because according to Wikipedia the supernatural is that which exists but is not nature. What is a golp? The opposite of a ferk. But what is a ferk? The opposite of a golp. Fun, isn't it? At some point one would have to define at least one of the two concepts with reference so something else.
So those are the two definitions of supernatural offered by Wikipedia, and I fail to understand the utility of the concept. There are really only three possibilities: if something has some observable effect on the universe and follows some rules/laws/logic, then science can get to work. If it does not have any observable effect on the universe, then there is no reason to believe it exists (and if we are talking gods here it would also be pretty pointless to worship them). If it behaves in a completely capricious fashion, then nobody can make any claims about the supernatural and the point is moot. But again, the latter two possibilities are not what the believers mean anyway.
In practice, however, there is a third definition of supernatural:
(3) that which scientists should keep their hands off because I say so, thank you very much
It is in this question-begging way that "supernatural" is all too often used in public discussion. When some philosopher, theologian or accommodationist says that science cannot address the claims of religion in general or the existence of gods specifically, they generally do not make any attempt to actually justify the supposed inability of science to do so, and they generally do not examine a specific claim made by religion to show precisely how and why it cannot be examined scientifically. The argument, if it deserves to be called that, at best boils down to saying that science is a priori committed to methodological naturalism.
But what does that mean? In simple terms, I think the point is really that scientists should not invoke spurious science-stopping explanations. It is about a scientist saying, "I think what I observe is done by demons, so now I will pack up my things and go home." But is it really about a scientist saying, "we have explained many things to satisfaction without invoking demons, there is no evidence that demons exist, and I will now, as always of course tentatively, conclude that they do not in fact exist"? It seems to me that the former is what methodological naturalism is meant to avoid, but the latter is quite simply a logical conclusion following from what science has found out so far.
But I don't even believe that the former, the science-stopper argument, makes a lot of sense. Scientists would not pack up their things and go home when faced with a demon. If we were living in a universe with souls, scientists would be happily studying their characteristics. If we were living in a universe with more communicative deities, theologists would simply be scientists and could find out some reliable and universally agreed on facts about those deities. If we were living in a universe where the spirits of the dead could be summoned into a magic circle and asked for information, every university would have a department of necromancy to provide this service to historians, just as they now have electron microscopes as a service to those who need them. That there are no real scientists doing any of this is because it very much appears as if there are no souls, deities or spirits, not because they are afraid of some scary philosopher of science who has told them that they are not allowed to do it.
Again, if it is observable and shows some regularity of behavior, it is just another part of nature that somebody tries to fence off by definition. If it is completely chaotic, it is not what anybody means anyway. And if it is not observable, then why is the scientist not allowed to conclude it does not exist? I sincerely see no reasonable answer to that question. "Supernatural" cannot merely be a word that a priest can arbitrarily slap onto some random claim that popped into their head to ward off the principle of parsimony. To me, allowing the scientist to conclude, based on absence of evidence, that phlogiston does not exist but forbidding them to conclude, based on absence of evidence, that immaterial souls don't exist, appears to be quite simply special pleading and intellectual inconsistency.
(In practice, of course, most scientists have better things to do with their time than running around and publicly concluding something like that unless driven to exasperation by religious intrusion into the political sphere or similar, and in practice a minority of religious scientists would disagree. In practice, I myself am not working in a field where I would have anything to do with the entire issue unless faced with creationists. This post is merely an entirely academic exercise in pondering the topic of methodological naturalism.)