Well, as an asian, I find that Coles is the best place to find asian food, ingredients to cook asian meal besides some asian groceries which are not in my area of living. Woolworths tends to be a bit more expensive while Aldi is cheaper with limited selections. I remember the time when I went shopping at Safeway, IGA but now they have fewer places than Coles and Woolworths which are spreading out to everywhere. I would love to go shopping at the markets if there are any nearby but sometimes it's difficult to find some asian items in local markets.
I think it's difficult to say what the best supermarket is, it depends on many factors, so you just pick what is the best for you, that's it.

The best asian grocery I've ever encountered is the one at Melbourne Central next to Coles but it would take me more than an hour to get there from where I live so it's not for daily shopping but just, perhaps, monthly.

I agree DrGunner. We eat a lot of asian foods as well; some sourced at a chinese grocer in the CBD near where my husband works.