The Boise Public Library offers 3 free downloads of top-40 type songs every week. That is, every week you can download three DRM-free songs. What you need to sign up is a library card number for the Boise Public system.
The system is called Freegal. I looked to see if this existed for my local library after I read a piece (I cannot remember, unhelpfully, where) about other libraries that purchased rights to MP3’s for people to download.
The content is very limited – they are largely top 40 albums and songs. I was able to find albums for Broken Bells, an alternative band I like, but that’s likely because they’ve broken into the mainstream charts. The Shins were there as well. Led Zeppelin is not. There are two albums of Mozart. Taylor Swift came up with a few covers. But 3 downloads a week is better than nothing, and it’s a good start to alternative media distribution models in libraries.
I found this by digging around in the “digital media” and “e-content” part of the website – if you want to find out if your library has something along the same lines, that’s where you’re going to want to start. Because libraries are investing in this, they are likely going to try to publicize it as much as possible so that patrons get use out of it. If you can’t find it on your library website, connect with a librarian and see what is available.
Related links: Lifehacker’s guide to How to (Legally) Boost Your Music Library Without Spending a Dime
Addendum 3-17-2014: Heard from someone in West Lafayette, Indiana – they have a program like this too.