'Round these parts, while we do drink a bit while we sing (normally between tunes), we dont sing too much while we drink (too complex a task) - if we do it we dont usually remember doing it let alone what we sang.   We'll sing about upcoming drinking, then drink, and then sing later(usually bemoaning our condition). As such these arent really traditional "drinking songs" so much as songs about drinking
- but some have potential for good sing alongs:

Bottle of Wine (Tom Paxton)
Mountain Dew
Rye Whiskey (Drunken Hiccups/Jack of Diamonds etc)

I always kind of liked the Pig Got Up and Slowly Walked Away, but havent done it for years.

Of course in the Fifth Grade I recall singing "99 Bottle of Beer on the Wall" just to annoy my teachers.

And there are the recent country hits - Friends in Low Places and I Love this Bar