Can PhD students publish papers as sole author without including their supervisor?

If the supervisor does not actually write anything or contribute in the paper, can a PhD student write as a single Author?

Yes. If for no other reason than I did, which suggest its possible

Would this be a faux pas and cause a strained relationship with the supervisor?

It depends on many, many factors. Is this a side-project that's been taking away from your actual work? Then it probably will. Have you talked to your supervisor and decided that it's the right way forward? Then it probably won't.

I know they expect their name to be on all papers related to the PhD.

This is not universally true. While I expect to be an author on most of my graduate student's papers, that's because I expect to have done work on most of them. With an upper, and potentially likely bound of "all of them". But if a student wrote something independent, then no, I don't expect my name on it.

Also, does using their lab mean they have a right to have their name on the paper?

It depends on what you mean by "using their lab". Generally speaking, just providing equipment isn't enough to warrant authorship, but if I did use someone's lab, I'd take a long look at the "did nothing" assumption and make sure it's true.

That's also potentially the source of a strained relationship - if you're using my space, equipment, and potentially reagents for a project I'm not participating in, I'd be less than thrilled unless it was discussed ahead of time.

Generally speaking, providing funding/equipment/space does not warrant authorship. Authorship is granted based on intellectual contributions to the work. It is somewhat rare for a student to conduct research and author a paper in which his/her supervisor contributes nothing, but if that were true, you could make a case for single authorship. But, such a proposal can be a faux pas, if it suggests you do not appreciate or even acknowledge your mentor's input.

Short answer:

Yes, if the supervisor has not contributed to the scientific work.

No, if the supervisor provided the specific question or direction of research.

And, yes, it may cause some resentment from the supervisor. Though this depends on the supervisor, the area, and the past work and relationship between you two. In your case, you explain precisely why it will cause such a resentment.

My conclusion then: unless you have an extremely strong reason not to, you should consider including the supervisor as it does not detract in any significant way your credit.