I have two. Audiogalaxy and Google Music.
I use an Android phone, and both application/services have native player clients that run on the phone, and allow over the air syncing to the device. These applications allow me full access to my entire personal music collection anywhere at any time.
Both applications have very good browser based players.
Both applications are free (except for the ability to sync music locally with Audiogalaxy. That is a pay feature of their phone application).
Google Music allows the free storage of up to 21,000 of your music files in virtually any digital form, including lossless codecs. The advantage of Google Music is that all of your files are stored on their servers, and you are able to re-download your own music to your own computer. Plus, as I mentioned before, the native phone client allows you to sync music to your phone for offline listening, if you choose, over the air with no need to plug your phone in to your computer.
Audiogalaxy does not have any limit on the number of music files you can store, but your full storage is on your own computer hard drive. Audiogalaxy has a "helper" service that runs on your computer. This helper allows you to log in to your own account through the native phone app (iPhone or Android) and access your own music collection over the Internet. The downsides are that although the app is free, you have to pay for the add-on that allows you to store music to your phone for offline listening (this doesn't matter on an unlimited data plan or over wifi), and your personal computer must be on and connected to the Internet at all times.
Both applications allow you to use the browser interface anywhere with an Internet connection, and both applications serve up the full quality of the stored file.
If you're talking for recording/mixing, I'm not a pro, and my tastes are simple. Guitar Player Pro for learning and composing new work, and Cakewalk for recording and mixing.