Hi, Michael - I've added links up top to our other threads on the song. As you can see, there is disagreement...
I believe siúil a rúin is the exact spelling, if you can use fadas. I prefer "Shule Aroon," because it aggravates the pedants.
Who's the traditional American singer who sings it "shool a rack-shack, shool a barbecue"?
-Joe Offer-