There are plenty of companies offering subscription music services in Australia, but virtually all of them ask for a monthly fee. Spotify is one of the few that offers a free ad-supported service, but you still have to pay to use it on a mobile device. Enter Guvera: An Australian developed free music streaming (supported by advertising) which you can access on iOS and Android devices.
You can sign into Guvera using Facebook or set up an account via your email address. The site claims “millions” of tracks; my searches suggest that the range isn’t as broad as Spotify and there are some problems with metadata on quite a few tracks, but there’s enough there to keep your playlist ticking over for quite a while. If you want to avoid the ads, you can sign up for an $11.99 a month Platinum account. The free option will probably be enough to please casual listeners; as with any streaming service, stick to Wi-Fi to avoid excessive data charges.