Wild Mountain Thyme sung by Connie Dover (that one might start an origins argument)
Craggen White Hare sung by Andy Irvine
Lakes of Ponchartrain by Paul Brady (an American song)
Thousands are Sailing by the Pogues (The newer version written by Phillip Chevron)
Little Musgrave by Planxty
Marie's Wedding
Cragie Hill sung by Dick Gaughan (is that one Irish).
Pair of Brown Eyes by The Pogues

so many more great ones...