Adding to this discussion, since my Irish-theme gigs will get into full swing about next Wednesday, I'd say the list depends a lot on the venue. When I sing for retirement communities (in the US), the folks want to hear the 'Irish' songs they know and love (most of which are Irish-American), e.g.:
Too-Ra-Loo-Ra-Loo-Ral
MacNamara's Band
My Wild Irish Rose (with both verses)
When Irish Eyes Are Smiling (with both verses)
The Kerry Dance
Tipperary
Who Threw The Overalls In Mistress Murphy's Chowder"
"Harrigan" (and the parody "Hair Again [On Me], which I dedicate to the "follically challenged" in the audience)
"Cockles And Mussels"
"A Little Bit of Heaven Fell"
"Peg O' My Heart"
"Peggy O'Neil"
"Mother Machree"
"No Irish Need Apply"
and, of course,
"Danny Boy" and
"When Irish Eyes Are Smiling"

I usually throw in a couple of show tunes like
"Great Day For The Irish" and
"How Are Things In Glocca Morra"
as well as some of MY Favorites, such as:
Roddy McCorley
Mountains o' Mourne
Shule Aroon
Fennario
Maggie
If I Knock The "L" Out Of "Kelly"BR> Johnny Lad
The Minstrel Boy
Finnegan's Wake
Hares On The Mountain
Wild Mountain Thyme
Black Velvet Band
Song For The Mira
Come Back, Paddy Reilly
Blow The Candles Out
and, yes, The Wild Rover
Whiskey In the Jar
The younger the audience is, the more I steer towards the songs in the latter part of my list.

If my audience has diminished attention span (either due to Alzheimer's or alcohol), so that they're unlikely to track the lyrics, I may skip some of my favorite humorous songs such as:
To Morrow (the TUNE sounds Irish, anyway)
The Sick Note
No Irish Need Apply
Nell Flaherty's Drake
Quare Bungle Rye
Mick McGuire
Twa Heids Are Better Than Yin*
Ye Canna Shove Yer Granny Off A Bus*
The Orange And The Green**
Three Craws*

Many of the songs that would be very popular in Ireland or for the US folkies who are "into Irish music" may not be what the "piper who is paying me" has in mind for a St. Patrick's party. It never hurts to check out these expectations beforehand.

Anyway, to my delight, I have a much bigger repertoire of Irish and Irish-American (and other British Isles) music than I can fit into a typical gig. March is one of my favorite months, since it gives me an excuse for singing a lot of these songs.

Genie

*If a gig isn't right on St. Patrick's Day itself, I 'cheat' sometimes and throw in a Scots or Welsh song or two.
**I think I like this song especially because the story fits a bunch of my ancestors' situations.