The soundtrack is a delight!

I love the melding of melodies in the opening credits and the use of music throughout. But, the IMDb listing is far from complete. I'll start with the opening medley:
"Irish Folk Song Medley"
Traditional Irish music played over the opening credits include
"The Minstrel Boy"
"Irish Washerwoman"

I also heard these tunes:
St. Patrick's Day
Garryowen aka GarryOwen
Rocky Road to Dublin

Later, I heard fragments of these songs:
The Orange & the Green
Garryowen
The Minstrel Boy

Following "London Bridge", the track includes "British Grenadiers".

I'm going to keep listing, as I'm not quite at the half-hour mark, where Billie Burke (who was played by Myrna Loy in "The Great Ziegfeld" and who appears in that scene with her) is playing "Kathleen Mavourneen", or so they say. It doesn't sound like the song I've heard sung so often. Sometimes, Irish songs when hammered into less-traditional modes sound entirely different.

~~MystMoonstruck~~