In my completely simpleton and often wrong opinion, the risk to the banking system is not a bank run but a major fall in the property market.
Just as in the US, if property values fall by half then many borrowers will be underwater and the bank will have an enormous pile of impaired 'assets' on their books.
Only thing that makes it more likely Australian banks will survive over US banks is the lack of non-recourse loans in Aus (NR chose his username for a reason

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In the US if a loan on a property is underwater, the borrower can just hand back the keys and walk away from the loan.
In Australia if a bank forecloses on a mortgage and sells the house, if the proceeds of the sale are not enough to cover the loan, the bank can continue chasing the borrower to pay for the rest.
Read an Australian mortgage if you have one - if you have a home loan in arrears the bank has the right to grab money from any other accounts you may have with them.