There are 2 key weather patterns to look for for snowfalls in Australia, one is the Cold Outbreak and the other is the Cutoff system.
Usually responsible for dumping a lot of snow over a short period cold outbreaks are relatively common and can even occur during summer. Cold outbreaks are short lived fast moving fronts that will dump snow for one or two days. They are often accompanied with high winds (the isobars are close together) and followed by about 5 or 6 clear sunny days.
Cold outbreaks are common early and late in the season. They will often drop lots of snow in Tassie, and southern Victoria and may not even reach the more northern resorts.
The key for a good season is a few good cut off low pressure systems. During summer the centre of the high pressure systems pass over Tasmania or Victoria resulting in easterly and north westerly winds. During winter the centre of the high pressure systems move north resulting in a stream of west or south westerly winds.
In certain conditions a low pressure system will feed cold air into the system and can result in snow falling for a week or more at a time. In this case the more northern resorts will often get a lot of snow while the more southern resorts can miss out!
Beyond the basic synoptic chart as shown on the TV or newspaper you can go into a lot more details such as the 540 line, the freeze level and a range of other parameters that indicate imminent snowfall.