When you say publishing, do you mean in a peer reviewed journal, or as like...a monograph through a publishing house or self-published?
For the former the answer is sort-of-yes. I highly doubt the full dissertation would be published as is. However it may be rewritten as a short article and submitted, either by yourself, with your supervisor, or more likely, as a publication primarily written by your supervisor but using part of your dissertation (data for instance) and with you as a (subsequent) named author.
Given the time scales involved in publishing in peer reviewed journals, this will normally happen well after you graduate. I do believe one of my friends wrote a paper with his (PhD) supervisor using his dissertation, which was published in the second year of his PhD. One of my other friend's PhD thesis directly continued on from his undergraduate dissertation (with the same supervisor) so inevitably he now has publications related to if not directly using his dissertation - during his PhD.
These examples are all in STEM fields though (also as I recall they all got high 2:1s or 1st class marks, at least overall if not in the dissertation itself).
For the latter I doubt it - I mean you could self-publish but that would just be...awkward. Perhaps if you became extraordinarily well known in your field there may be demand for it...maybe. In any case individual universities hold a copy of all theses and dissertations which are usually available for viewing by request, at least internally. So for that purpose it's a bit irrelevant.