First publication – author list better alone or with supervisor?
Unless there are more people despising your supervisor than respecting them, adding him to the publication will inevitably increase its visibility by some amount, even though this should not be the case in an ideal world. There is a positive chance that somebody will only read your paper because they know the author. How large this effect is depends on your particular subfields culture and the popularity of your supervisor. Moreover, note that your supervisor can also advertise your work to their peers irrespective of authorship and this effect may be more relevant to the impact than authorship.

That being said, for your imminent career, it is mostly irrelevant how many people actually read your paper, but only how relevant or good the paper is considered by those who decide upon your applications. However, those people will also know that you worked with your current supervisor and will also likely have a recommendation letter from them praising your work in the paper. So adding your supervisor as an author shouldn’t matter in this respect. On the contrary: It’s certainly more impressive that you single-authored a paper.

Now, for your later academic career, you have to weigh between the two effects, but since you will hopefully have written several papers then (otherwise you will probably not have a later academic career) and securing a good PhD position may help for this, my bets are on opting for being a single author.

A sidenote: Going from your description, I assume that it is borderline whether your supervisor qualifies for authorship and thus ethical considerations of authorship are inconclusive.