There have been a lot of good suggestions so far. You could fill the half-hour more than a dozen times! Of course, with only a half hour you can offer only the trace of a taste--and I suppose that's what's intended. All the genre have been covered pretty well, but no one has mentioned the variety of stylistic traditions that existed (and still exist) in Ireland.

I have given a similar workshop on several occasions (although I've given myself 45 minutes to an hour) and have used the following format:

I first point out that the Celtic tribes were noted for their singing as far back as Heroditus (I think it was he), who wrote (in the fourth century BC) that they sang and chanted before they went into battle.

I play a recording of one of the last singers of the Ossianic Lays, the oldest Irish song type known today. As I remember it, the last of these singers died in Scotland in the 1930's. These were ancient chants about Celtic mythology. I believe I got my recording from Virginia Blankenhorn when she helped teach the singing class at the Willie Week many years ago.

Then keening. (I would attempt the Irish spelling but I know I'd make a haims of it.) This tradition of singing is performed almost entirely by women and is, as I understand, still practiced by a few people. It might be even more ancient that the Lays.

(These first two examples would be very brief.)

Then songs in Irish sung in the sean nos (or old style). A number of them have been mentioned in this thread. A classic sean nos song I don't remember being mentioned yet is Rosin Dubh.

Anglo-Irish songs in the sean nos. Many to choose from here—from the singing of Paddy Tunny, Len Graham, Joe Heany, Tom Lenihan, and many others.

Classic (Child) ballads (only a few in the Irish tradition)

Broadside ballads (a rich tradition here)

Parlor songs--usually looked down upon by folkies as sentimental slop. However, it has been one of the cores of the Irish singing tradition with "Moore's Melodies" leading the way. It has been said (with probably only a little exaggeration) that in the last part of the nineteenth century almost every Irish household had a copy of Moore's Melodies. Any sampler of Irish songs, especially for an Irish studies class, would be remiss not to include a Thomas Moore song.

Music Hall songs and other comic songs. Gotta include a Percy French song.

Although rebel songs constitute more a genre than a style, there is a rich tradition there, too.

Then, if you want to go there, newly composed songs like the Town I Loved So Well or Four Green Fields.

So many Irish singing traditions.

Half and hour? Oh my. Good luck and have fun!

David