Like other reluctant federations, Australia has a constitution that safeguards powers of the states. These have become less attention-grabbing with time; Australians are still parochial about their states, but they’re more excited when that parochialism involves beer brands than legislature.
The WWII switcheroo, whereby the Federal Government appropriated all taxation power of the states as an “emergency” measure (not repealed, unsurprisingly) has given the states a lot less heft anyway. And most migrants to Australia, my parents included, scratch their heads at federalism, and ask “why do they keep the states around?”
The states do run a lot of day-to-day stuff that isn’t as visible as federal politics, but is likely more pervasive. And the states would in fact be within their rights to adopt an established religion, and do all sorts of other things the constitution doesn’t enumerate. But we have not had much of a discourse of states’ rights. The most visible it gets is the annual haggling over money handed out from Canberra (the Feds hand back to the States the money they can’t raise on their own).
The closest we have is resentment at the centre, most overt in the states furthest from Canberra, Queensland and Western Australia, which maps to states not wanting to be told where their money goes, or how to run their affairs.
Regional Queensland politicians, for example, have tried to cast Marriage Equality as Federal Interference—which I guess is the closest to States Rights US-style. You could argue that those politicians are at odds with the population of their own state capital. To which they will answer that if they had their way, they’d secede from Brisbane anyway. The ACT, on the opposite side, passed marriage equality legislation, which was struck down by the High Court—because over here, marriage is a federal issue, not a state/territory issue.
But no, not highly visible.
(I suspect I’ve got a lot of this wrong, and I request Tracey Bryan to post her own more correct answer.)