I advise clarifying with the supervisor whether "go ahead and publish it yourself because I cannot contribute more to it" meant that you should submit it with only your name in the author list, or whether they meant that in their opinion it was ready for submission and that they had no suggestions for improvement.
At the University where I completed my PhD, and at others Universities of which I have direct knowledge, it was the norm (i.e. expected) to include supervisors as co-authors independent of direct contribution to a paper.
In the case where you are the (otherwise) sole author then the author list would read < self >, < supervisor1 >, < supervisor2 >, ... . And in the case where you were the lead author and had contributions from others then the author list would read < self >, < co-contributor1 >, < co-contributor2 >, ..., < supervisor1 >, < supervisor2 >... .
I have not been able to find anything which states this explicitly. I have since found this, which to me indicates that supervisors should only be listed as coauthors if they directly contribute, contradicting what was communicated verbally to me:
List the authors on the title page by full names whenever possible. Please be absolutely sure you have spelled your coauthors’ names correctly. Be sure also to use the form of the names that your coauthors prefer. Include only those who take intellectual responsibility for the work being reported, and exclude those who have been involved only peripherally. The author list should not be used in lieu of an acknowledgments section.
(Additional Information for BSc Honours Dissertation and MSc Research Projects)
On reflection, this suggests that early papers will normally have supervisors as coauthors (assuming intellectual or tangible contributions to the papers) as the PhD candidate develops as a researcher, but that later papers would only list supervisors if the supervisor has clearly contributed in the same way as would be considered for any other person to be listed as a coauthor.
(Independent of this, I have also seen academic environments where it is politically expedient to list supervisors as coauthors - to keep them happy, so to speak...)