What is the scientific quest for God?
First of all, it is helpful to understand what the scientific quest is not. It is not, or at least it should not be thought of as an attempt to prove that God exists. No one expects to find him experimentally. No one expects to see him staring back at us through the Hubble telescope or some probe into space. He doesn't figure into our great formulas such as E=mc2. The God of Christianity and the other Abrahamic faiths is not a natural law. He is not a theory. He is not a hypothesis.
Secondly, it is important to recognize that the scientific quest for God is really is an apologetic: a defense of a belief in and about God in response to claims that he does not exist or need not exist as evidenced by science. Also, it must be realized that the scientific quest it isn’t completely scientific. In addition to real and true natural science it includes, in some aspects, what most scientists and philosophers of science think of as pseudoscience. As much as this may be unattractive to some of us, we must confront it because many capable scientists, and much of the non-scientist population at large, thinks of the pseudoscientific component, primarily something called Intelligent Design, as real science.
The primary categories of the scientific quest for God are: 1) Evolutionary Intelligent Design (ID) and 2) the apparent Fine Tuning of the Universe. The first of these, in response primarily to Charles Darwin's Theory of Evolution, is widely considered pseudoscience and second is not. Interestingly, most proponents of ID also embrace fine tuning but serious scientifically-minded proponents of the fine-tuned universe arguments do not generally embrace the evolutionary ID arguments.
Other categories of scientific quest for God (or for proof that there is no god) are found in neuroscience, quantum physics, string theory, and abiogenesis. If we are to accept what Philip Ball thinks, then the Shroud of Turin is also a category.
Academically, the Shroud of Turin, in this context does not get as much attention as other categories. But in the media it does. Thus, to some extent, the public, at large, at least those who follow developments pertaining to the Shroud of Turin, sees it that way.
It should be noted that Creationism, sometimes called Creation Science, is not considered. It is, with some variations of interpretation, a rejection of scientific findings in favor of a literal or nuanced-literal interpretation of the Genesis from the Old Testament of the Bible. It simply does not qualify as science.