Don't know about songs but tunes, Some of the favourites that I guess get played everywhere are:

The Merry Blacksmith, St Annes Reel, Harvest Home, The Boys of Bluehill, Morrisons Jig, The Kesh Jig, Lark in the Morning, The Silver Spear (not to be confused with the Silver Spire which is also poular), Banish Misfortune, pick on a slow one for the last of this 10 - Planxty Irwin. The list may vary from day to day but I think this selection will be pretty well known by most musicians who play in Irish type sessions.