When riding with a single pannier, does it matter which side it's on?
I wasn't aware of a rule for this, but I would hang a single pannier on left because it's the non-drive side of the bike. I doubt it matters though.

When I was buying a new rear wheel a few months back, I found the spoke patterns in some rear wheels are different on one side to the other, so I don't know if this would make a difference.

(I'd have added this as a comment, but it informs me I'd need 50 rep for that, hence the "answer".)