Well, I know about 75 sea chanteys that are exclusively considered to be drinking songs. Of these, 38 mention Australia, but I only know of 16 of them as being strictly Australian in origin.

Bound Away for Australia is a popular favorite around these parts, since it lets you get really raucous. There's also a whole class of songs that mention being sent (that is, exiled) to Australia, and most of these are lots of fun.

One favorite of mine is Maggie May about a young lady who "gets her passage paid to Botany Bay" for robbing so many sailors:

Maggie, Maggie May
They have taken you away
To toil upon Van Diemen's cruel shore
For you've robbed many a whaler
And many a drunken sailor
Now you'll never cruise 'round Liverpool anymore