A feature in the Advertiser on 30 May 1998 listed 100 Adelaide postcodes ranked by how good they are to live in. The bottom 5 (excluding 5094 and 5106, which are not really residential) are right in the middle, some of them fairly rich.
I don't remember the criteria. There is an obvious bias against the inner suburbs, so ease of getting to the city appears not to have been a significant criterion.