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.