Music to know

[Freddo]

"Be Thou My Vision"
"Glenswilly"
"Spancil Hill"
"Wild Colonial Boy"
"O Sullivans's March"

[Wormdiet]

For "recognizable" you could mine the Ken Burns documenary about the Civil War.

I think I have stumbled onto the perfect jig: The Monument

[LeeMarsh]

the most recognizable jig for most non-irish is:
The Irish Washer Woman.

Non-irish tunes that play well on whistle, especially low whistle is

• The House of the Rising Sun.
• Scarborough Fair.
• All Through The Night.

As for slow airs/song, I'd suggest:

• The Month of January (use Doloris Keane's or John Doyle's recording as a guide).
• Culloden's Harvest (Deante has a nice recording of this song).

A nice waltz is Far Away.

Hope this helps you find more ways to ...