> I have never heard the word "wain" used on its own.

In isolation, Charles's Wain, also known as the Plough, the Great Bear, the Big Dipper, and probably quite a few others. I know I should call it the Plough but I always say Charles's Wain.

In compound: a wainwright. (And does Tolkien use Wainrider somewhere? Who were they... some warlike tribe from Rhun?)