The first is a paper by Fishman and Boudry that makes the point that methodological naturalism is not an a priori commitment of the scientific enterprise; rather, it is a conclusion arrived at in the same way as other scientific results, namely, by applying the usual criteria of economical explanation of phenomena. They argue that "supernatural," as it is commonly used, is too loose a term to be useful. Instead, they say, we should talk about the "overnatural" - beings or powers that are similar to natural ones, but beyond what is normally possible (like Superman or walking on water) - and the "transnatural" - things that are "categorically different from ‘natural’ ones, so much so that their properties are essentially mysterious, ineffable, and incomprehensible."
The overnatural can be investigated the way we investigate any scientific hypothesis. The transnatural, on the other hand, cannot be scientifically investigated, since it (by definition) is incomprehensible and inexplicable. However, they argue, the transnatural is an empty concept. They quote Martin Mahner:
There neither is an ontological theory proper of the transnatural nor could there be, because there can be no theory of the unintelligible.Scientists thus reject a transnatural explanation, not because of any a priori commitment, but for the same reason they reject any other unintelligible or ill-formed hypothesis.
The second bit is a blog post by Victor Reppert, who argues that the supernatural is a claim that minds are not composed of non-mental things - they are part of the "rock bottom level of the universe." This is really a thesis about dualism rather than a definition of the supernatural. But I think Reppert is right to focus on minds in talking about the supernatural.
Imagine that large boulders spontaneously levitated at random times. Such a phenomenon would be inexplicable according to current science, but I don't think it would be considered supernatural. It would just be another kind of natural phenomenon to be described. On the other hand, suppose they only levitated when a car was about to crash into them. Then we would suspect a purpose behind the levitations, and a mind behind that purpose. Since the phenomenon is beyond the ability of any human, we would have to suspect a supernatural being. (Or, I suppose, a powerful extra-terrestrial who can know about these events in advance and who has the ability to move rocks from a distance. In effect, such a being would be supernatural from our point of view.)
This all ties in to the discussion about the fine tuning argument for naturalism, and what sort of observations would be considered evidence of supernatural intervention - but I'm still thinking about how.