I was going to post Frank Sinatra singing an Irish song…but I could not find any. Sinatra apparently liked hanging out with the kitchen staff at Dickie O'Neal's Irish Pub, so we have that.
And here is a post on Frank's Irish Connection! (and it does have a song)
Mark Steyn: I've Got You Under My Skin (Porter Riddle Sinatra Synergy), Sinatra's Other Pursuits (film), and South of the Border (Mark had the same trouble I did when it came to Irish songs).
Bob Belvedere/Camp of the Saints: Sinatra's Best 82 through 80, 79-77, 76-74
Pundett: #81 Hooked: You're getting to be a habit with me… and #80 Cheek to Cheek
If I have to pick a song, why not this song that was done by an Irish band too:
I''ll throw this in too: