I hate it when people say "Southern Irish". We're not southern, we're just the rest of the island. I mean, Donegal is more geographically northern than Northern Ireland (which is more like North-East Ireland), and you're saying it's "south"? It just don't make no sense.
Also, Irish Americans who don't qualify the half-American part can be pretty annoying too: "My great-great-great grandma lived for a couple years on Leprechaun Isle! Which makes me Irish! Which means I can make all the stupid Irishmen jokes I want without offending anyone!"
And of course, anyone who refers to the Irish as British get beat up. That's a given.
Though it doesn't bother me as much when Northerners call themselves Irish (although it'd be less confusing if they didn't. You want to be a separate country so damn much, you have to think up your own separate collective noun!)
The Irish are generally a nation of moany narcissists who resent other cultures exploiting how awesome it is being Irish. We work hard for our reputation as slick, witty, drunken party-animals/bar-brawlers. We've got like, the highest per household rates of alcohol-related deaths in the world! We pay the price for our coolness factor!