3rd March 1986 is the date the Australia Acts were enacted, which is actually what removed the last shackles of UK dominion over Australia, including the UK's ability to pass legislation that would have effect in Australia or for Australians to appeal to the UK privy council. Up until this point, Australia was officially still a dominion of the British Empire.

The Statute of Westminster was officially adopted on 9th October 1942, though it was backdated to 3rd September 1939. It removed most of the ability of the UK to legislate with effect in Australia (still enabling them to legislate over Australia with Australia's "express request and consent"), removed the ability of the monarch to refuse to allow the Governor-General to give royal assent, and gave Australia the ability to control the laws of royal succession independently of the UK, which is why prior to William having his first kid, Australia (and other countries such as NZ, in addition to the UK itself) had to enact legal changes to the laws of succession in order to allow for succession to be simple primogeniture, rather than male-preference primogeniture.

So that's 3 good dates there: 3 March, 3 September, and 9 October.

There's also 9 July, which is the date the Australia Constitution Act 1900 was enacted by the UK Parliament, allowing for federation to occur on 1 January the following year. So a potential 4th date, but kinda a lame one IMO.