We'll be spending Christmas in Vienna. Anyone with any ideas about the best place to hear some Christmas music? Will there be choirs singing at Christmas markets or are there any other public events?

Concerts are one option, another will be church services. Most churches do festive services with special music during the holidays. Check out what's on in Stephansdom...

To be honest, it's harder to escape those concerts than finding them.
This site* will give you just an idea, and there will be much more smaller concerts (which have more flair than the big ones that attract mainly tourists).
*) select location = Austria and Genre = Christmas / New Year and a date range.
Unfortunately you can't select Vienna only, so you have to scroll to find the concerts that will be played in Vienna.