Did Judy Garland lip sync "Somewhere Over the Rainbow" in the "Wizard of OZ" having recorded it in a studio prior?

What a good question! The vast majority of movie musicals did (and still do) prefer to get a perfect recording in the studio, and then for the actor to mime (or lip sync as you say in the US) when recording the visuals. They've all done this. From Sinatra to Streiand. From The Sound of Music to Chicago. One notable exception is the finale of Moulin Rouge. At the end of that movie, McGregor and Kidman do sing live. But for the rest of the movie, it's miming. Some movies, like West Side Story and The King and I, went further. The lead actress is miming over someone else's singing!

I can't find any actual proof though that Garland mimed her Over The Rainbow part. I'll need a bit longer with this one.

The extra features of a fairly new DVD of "Fiddler on the Roof" goes into detail on the recording and shooting of Reb Tevya's famous "If I were a Rich Man" scene. It was all lip-synced and took something like three days to put together. The time was well spent!

I can't specifically confirm that was the case in "The Wizard of Oz," but I can't imagine it wasn't.

OK, here it is, "The songs were recorded in a studio prior to filming."

https://en.wikipedia.org/wiki/Musical_selections_in_The_Wizard_of_Oz