As regards actual songs ABOUT St. Pat, there's this.
Don.

SAINT PATRICK WAS A GENTLEMAN

Saint Patrick was a gentleman and came of decent people
He built a church in Dublin town and on it put a steeple;
His mother was a Callaghan, his father was a Brady
His sister was an O'Houlihan and his brother an O'Grady.

CHORUS: Success attend Saint Patrick's fist, for he's a decent saint O
He gave the bugs and toads a twist; he's a beauty without paint O.

The Wicklow hills are very high, and so's the hill of Howth O
But I know a hill that's twice as high and latter than them both O;
'Twas on top of that high mount where Patrick preached his sermon
He made the frogs jump through the bogs and he banished all the vermin.

No wonder that we Irish boys should be so gay and frisky,
For Saint Patrick taught the happy knack of drinking of the whiskey;
'Twas he that brewed the best of malt and understood distilling,
His mother kept a shebeen shop in the town of Inniskillen.

Then should I be so fortunate as to go back to Munster,
Och, I'll be bound that from that ground again I would not stir;
'Twas there Saint Patrick planted turf and plenty of the praties,
With pigs galore, a gradh mo stór (1), and buttermilk and praties.

(1) My love, my treasure.