There are, in general, two hypotheses about how the Shroud came to be. The first is that the shroud represents the work of human ingenuity. The second is that the shroud represents an artifact of supernatural activity.
We'll explore the supernatural hypothesis first. In very general terms, if something is the artifact of a supernatural process, we have no particular expectations about what sort of physical evidence we should expect to accompany it. In other words, there is no scientific way to test a supernatural hypothesis. The shroud could be the artifact of a supernatural process, and there is no way that this hypothesis could be completely ruled out, because it is not as though supernatural activity would leave any tell-tale marks.
So, there is really no scientific way to test the supernatural hypothesis.
Let's assume for a second that the shroud is an artifact of supernatural activity. If it is, we still need to keep in mind that there are at least two sources of supernatural activity. On the one hand, there is divine supernatural activity (including the various angels and wonderworking prophets). On the other hand, there is demonic supernatural activity.
There is, again, no way to discern "scientifically" the difference between the two kinds. Moreover, in this case there is no alleged witness to the shroud's creation. So, the difference cannot be discerned from the historical context.
So, again there is really no scientific way to test the hypothesis that the supernatural activity is divine.
Even if the shroud is an artifact of supernatural activity, and even if it is the result of divine activity, that still leaves the question: of what activity? Of course, Shroud lovers think that the activity has something to do with the person of Christ. Yet there is nothing on the Shroud that say, "Christ was here." There have been lots of people crucified. The apostle Peter is one example of other people that we know had supernatural power from God and who were crucified.
Yet again there is really no scientific way to test the hypothesis that the divine supernatural activity was Jesus as opposed to Peter as opposed to anyone else.
In short, even if the supernatural hypothesis is true, there is really no way to prove it scientifically, no way scientifically to distinguish divine from demonic supernatural activity, and no scientific way to discriminate between Peter and Jesus.
What about a natural hypothesis? If it is merely a natural artifact, then various hypotheses can be tested. However, of course, while the shroud is still interesting as an example, perhaps, of the earliest known pair of photographs. If that is what the shroud is, the technological evidence argues for a relatively late date.
No one that I know of has suggested that the shroud was created by supernatural power ex nihilo. Thus, a lot of the hypotheses are really mixed hypotheses: partly natural, partly supernatural. The natural parts of these can be tested. For example, the linen of the shroud can be tested by radioactive testing. It has been tested, and it came back as 13-14th century linen.
This has led shroud defenders to claim that the portion tested was a part of the shroud that was a repair to the original shroud. Thus, on this theory the shroud is only partly fake. To test this theory, it would be necessary to do radioactive testing on other parts of the shroud.
But ultimately, why bother? There is no real reason to suppose that the shroud is of supernatural origin. There is no possible scientific evidence that it is of supernatural origin, that it is of divine origin, or that it relates to Jesus Christ as opposed to someone else. At present, the most reliable dating technique has dated the linen to the 13-14th century. Shroud defenders will doubtless point to various criticisms of the testing, but at most those suggest that it would be nice to confirm the original results with more testing.
P.S. Jason Engwer has a post up on the question, "Is the Shroud of Turin Demonic?" Jason states that "we use probability judgments" about these things. But, of course, these are not probabilities in any rigor sense of the term. They are just speculation, and the "probability" ends up being fully determined by how much the evaluator wants a particular conclusion to be true.
There is one item of interest in his post, however: "The burden of proof is on the shoulders of those who would want us to think the Shroud is something other than what it seems to be." What it seems to be is a photo negative of a man, with blood applied to it, on medieval linen. The burden is on folks like Engwer to establish that it is more than it seems to be.
But, of course, he cannot meet that burden. It's not his fault - there is just no Scriptural or other historical support for the Shroud.
The most that scientific evidence could ever do for the shroud (in the best case for someone like Engwer) is to show that it was made in the 1st century in Palestine and has the blood of someone Jewish on it. In which case, we would still not actually know which of the myriad of crucified 1st century Jews the shroud pertained to.
In short, the shroud would be (by such tests) elevated to the status of the "Jesus tomb" ossuaries. But like the Jesus tomb ossuaries, there would still be nothing to positively connect the shroud to Jesus. It would just be wishful thinking on the part of shroud admirers - just as it is wishful thinking for the ossuary advocates.
Nevertheless, shroud advocates are much worse off than ossuary advocates. The shroud has already been proven to be from the 13-14th century at the earliest - the ossuary may well be from the 1st century.