With plans to take a chunk of the profitable digital music download market currently dominated by Apple’s iTunes service, Google Music has opened its doors today to US customers.

The beta phase has been running for a while, allowing users to upload their music collections and save them in the cloud so they can stream them from any device, but the new service now allows customers to share and purchase music through the official online or Android store.

The store opens with more than 13 million tracks from artists on Universal Music Group, Sony Music Entertainment, EMI, and the global independent rights agency Merlin as well as over 1,000 independent labels.

You can purchase individual songs or entire albums right from your computer or your Android device and they’ll be added instantly to your Google Music library, accessible anywhere and synched automatically across all compatible platforms.

The top-tapping ‘Introducing Google Music video’

To celebrate the launch, Google Music is offering a number of exclusive musical tracks for free including:

• The Rolling Stones are offering an exclusive, never-before-released live concert album, Brussels Affair (Live, 1973), including a free single, “Dancing with Mr. D.” This is the first of six in an unreleased concert series that will be made available exclusively through Google Music over the coming months.

• Coldplay fans will find some original music that’s not available anywhere else: a free, live recording of “Every Teardrop Is A Waterfall”, a five-track live EP from their recent concert in Madrid and a remix of “Paradise” by Tiësto.

• Busta Rhymes’s first single from his upcoming album, Why Stop Now (feat. Chris Brown), is available for free.

• Shakira’s live EP from her recent concert in Paris and her new studio single, “Je L’Aime à Mourir” are both being offered up free.

• Pearl Jam are releasing a live album from their 9/11/11 concert in Toronto, free to Google Music users.

• Dave Matthews Band are offering up free albums from two live concerts, including new material from Live On Lakeside.

• Tiësto is offering up a new mix, “What Can We Do?” (feat. Anastacia), exclusively to Google Music users. •

Google Music is open in the US at market.android.com, and over the next few days it will roll out the music store to Android Market on devices running Android 2.2 and above.

There are still a few odds and ends to tie up before Google Music launches in the UK and Europe.