Some swear by Tuesdays and Wednesdays, but there are no universal rules that are going to get you a cheap seat all the time.
As well as the day of your flight, destination, time you fly and when you book are all factors that feed into the price you pay, and there's no getting away from the economic lynchpins of supply, demand and the level of competition between carriers on your chosen route.
There are a couple of flight search engines that do a good job of helping you track down the cheapest day on which to fly, and the standout is Matrix Airfare Search (matrix.itasoftware.com). This is the cheap flights site you've probably never heard of.
Matrix Airfare Search has all the aesthetic appeal of a parking lot but it cuts to the nitty-gritty like a scalpel. Type in your route, the extreme ends of the dates for your outward and return flights, check the "See calendar of lowest fares" box under "Dates" and back comes a graphic with the cheapest fares available on every day within your time frame.
Click on your chosen day and as well as the airline with the cheapest fare you get multiple options for travel on that date with other airlines, since the cheapest fare might involve multiple stops, or possibly long layovers.
Along with almost every other flight search engine however, budget airlines with services to and from Australia such as Scoot and Air Asia X don't get a look in.