Don’t say “Danny Boy” now! 
No honestly, what kind of slow airs would you say any whistler can’t do without knowing?
None. There is no “should” regards to slow airs.
Just learn airs you really like.
And learn to play them well.
Most likely you play them solo anyway,
or with folks who showed them to you,
or with whom you play regularly
and who enjoy playing airs.
But of course there are some more commonly known airs,
like some from O’Carolan, which you might enjoy learning.
Most known of O’Carolan’s airs are perhaps
Si Bheag Si Mhor, Planxty Fanny Power, Planxty Irwin,
O’Carolan’s Welcome, Eleanor Plunkett, Hewlett.
Then some folks don’t regard O’Carolan’s compositions as true Irish,
as they show some continental baroque influence.
Some others (not O’Carolan’s):
The Wild Geese, Ned of the Hill, Give Me Your Hand.
I mostly play Scottish airs, especially from the Highlands and western islands.
But then i live in Scotland, and Scottish airs are more common here!
And i guess you were not asking for Scottish airs, but Irish 
.
~Hans
I don’t think it’s possible to answer this exact question. Unlike session tunes, which are a shared repertoire you need to know to play together, slow airs are solo pieces and are a personal choice. But I’m sure people will chime in with their favorite airs, without the idea of “should” attached.
The only airs I recall being asked to play by punters at sessions more than once are “Danny Boy / (London)derry Air” and “The Lonesome Boatman”. And technically, I guess the latter is really a “tune in the style of a slow air”, since I don’t think there is an underlying song.
Added: I see that Hans and I crossed posts.
In my opinion the compositions of Carolan are not “airs”, except by the very broad definition of “air” as meaning any sort of tune.
To me, “airs” are the song-airs, usually in the Irish language originally, the “sean nos” songs.
I suppose that Irish non-dance music could be broken down into categories
1) sean nos airs
2) tin-pan-alley quasi-Irish songs (Irish Eyes are Smiling etc etc)
3) compositions of Carolan and other old harpers (Give me your Hand etc etc)
4) pub songs/ folk music revival songs from the 1960’s etc (Fields of Athenry etc etc)
5) old songs in English such as broadside ballads, shanties, etc etc
well I don’t know if these categories would hold up to scrutiny…
Anyhow I’ve often found myself playing for fairly long periods as background music at weddings etc and I tend to rotate through these various genres in order to hit the various tastes of the audience.
I’ll play a New York quasi-Irish song, which is what most Americans think Irish music is.
Then I’ll follow with a sean nos air. The old Irish-raised people recognise those.
Then a folk music revival pub song which will be recognised by middle-aged Irish people and some Americans as well.
To me that’s the test of what airs we should know, at least if we’re being paid to play.
The airs that people seem to recognise and like are Carrickfergus, Rosin Dubh, and Laggan Love. Actually I learned Rosin Dubh because a few times when I was out playing it was requested by elderly Irish people.
The modern song that all Irish people seem to recognise and like is Fields of Athenry.
You’re right, Hans, there really is no “should”. The real question should ( 
) have been which are your personal favorite slow airs? I mean, the ones you often return to play, the ones you love to play.
Personally, I’m indifferent to whether they are Irish or Scottish - just do me a favor and mark them for which sort they are.
Here are a few airs which I think are lovely on whistle or flute:
The Parting of Friends
Bruach na Carraige Baine
Easter Snow
Dochas
Fath mo Bhuartha
Sliabh Geal gCua
Ta me 'mo Chodladh
–James
The accepted definition of “air” to traditional irish musicans is (generally, and simplistically) the melody of a song, or a tune played in that style. A “slow air” is generally accepted to be the definition of the tune of a sean nos or traditional (english language) song.
Your list doesn’t seem to take into account that the accepted definition of “sean nos” includes “irish language only”. English language songs in a similar style wouldn’t be considered sean nos.
Fields of Athenry is on par with Danny Boy!
The foinn malla I played to win last year at the Midwest Fleadh were “Green Fields of America”, “Valencia Harbour aka Amhran na Leabhair” and “A Stor Mo Chroi”.
Any slow tune in the categories above that makes you go “God, I love that melody!” is a candidate. There are even a few tunes in the category of “celtic new age crap” that regardless sound wonderful on a whistle :-p
huddles in a fetal position</flame retardant clothing>
“I love that melody!” Absolutely.
To James’ list I’d add
Bells of Shandon (better on harp than whistle) and
Sliabh na mBan.
Londonderry Air. I love jazz versions of it. I’m kind of kidding.
I hope I didn’t just make a derry air of myself. 
Nice way of not saying “Danny Boy”, anyway. 
Anybody else who’d like to tell me their favorite airs? - I’m off to google the ones you’ve mentioned above.
Yes, we need MORE!!
Well for as long as I have been trying to play O’Connall’s Lamentation, I would think that I should know it and play it right. Never have.
There’s always Blind Mary. One of my true hexes - I can never play it slow enough.
Blind Mary really sounds nice. - I’m definitely going to learn it by heart. Though I don’t think it’s going to be easy for me. There’s this one part that sounds very much like the middle part of “You are my sunshine”, a song I had to sing a lot back in 2001. So, everytime I get to this section my brain plays an evil trick on me making my hands continue playing the wrong song. 
I guess I’ll have to get the sheetmusic from thesession to solve this problem…
Sheet music to Carolan stuff is fine- it’s all we have (as the harping tradition was broken).
But the sean nos airs cannot be learned from the “white page”. Their timing cannot be forced into mathematical precision. The only way to learn those (and the best way to learn any Irish music) is to get a good recording of it and play along until you have it.
True. I don’t want to use dots to learn the timing or intonation. That would be useless because eventhough I know how to read them, anybody has ever achieved teaching me how to read timing and rhythm and the length of certain dots from a sheet.
When I try playing an unknown tune from a sheet of paper, I end up sounding like a first grader reading a book. While he or she knows all the letters, the letters often don’t form words or whole sentences in their brains so they stress the wrong syllables and nobody can understand their reading. - That’s me on any wind instrument I pick up.
When I still used to play the clarinet in earnest, I often bought or lent me a cassette or CD featuring the song I was supposed to learn. I listen to it in order to know how the melody was supposed to sound, or rather to know how long each dot had to be played…
Using sheet music to me is like using a visual anchor when difficulties like the ones I mentioned above arise. Seeing the dots and practicing that one part that my brain confuses with “You are my sunshine” is supposed to help me avoid that mistake instead of re-doing it again, again and again.
Yes, it’s the last bar of Blind Mary that matches “Please don’t take my sunshine away”.
Whenever someone starts Con Cassidy’s Jig in a session, I want to change immediately to “The Bear Came Over the Mountain”. Which leads into “The Teddy Bears’ Picnic”. Which is a great way to end up fleeing the pub, chased by angry pipers.