Chris Heard takes a stab at the question. His essay is a jewel of methodological rigor. He surveys the primary data with care rather than discuss previous solutions to the problem. Explanations of Selah in the literature are a dime a dozen, but none of them (the interpretations of the Targum and Jerome, Briggs, Mowinckel, and Eerdmans come to mind), so far as I can see, are convincing.
It’s nice to see someone conclude that the precise significance of Selah is a mystery. I refer the interested reader to Chris’s survey, and among the comments, the note that Selah also occurs in psalms and liturgical texts attested at Qumran which did not end up in the biblical Psalter transmitted by later Judaism. Selah also occurs twice in the Eighteen Benedictions, and is represented in the LXX Psalter in more instances than in the MT Psalter. The additional examples fit into the typology of occurrences known from MT delineated in Chris's essay.
Selah is, clearly enough, a musical rubric of some kind. This is implied by the oldest interpretation of the word on record, that of the LXX (Habakkuk and Psalms), which translates διάψαλμα. But διάψαλμα is a neologism in Greek. It refers to something musical, to what precisely is not obvious. It is not known to me on what basis Albert Pietersma takes διάψαλμα to mean an “interlude on strings” (NETS translation). It’s as good a guess as any, but I’m not sure it’s more than that.
If Selah is a musical rubric of some kind - a flourish, perhaps – that explains why it occurs at spaced intervals once, twice, or three times in a given psalm, and always in psalms that have superscripts plus or minus subscripts containing other musical information. It would also explain why Selah coincides with but does not indicate logical breaks (here I summarize the results of Chris’s research). If Selah were a discourse marker per se, we would expect to find it distributed in non-superscripted Psalms as well.