Babel Scores * is a new, but quickly growing online repository of contemporary scores that are (almost always) accompanied by recordings of the pieces and notes by the composer.
IMSLP (Petrucci Music Library) had over 275,000 freely available scores as of June, 2014. Scores available to Americans using this site have been published prior to 1923 (i.e., they're in the public domain). Generally you will find old 19th century editions of music that are good for a quick look, but are sometimes fraught with errors that more modern, scholarly editions have corrected.
Open Music Library helps you discover what music is out there on the internet. It regularly gathers information about new electronic sheet music, books, articles, etc. that are available online and provides a single search engine to find them. This is particularly helpful for locating small collections of electronic sheet music
Classical Scores Library * contains both public domain and in-copyright scores. This site is more valuable for the in-copyright scores it offers (generally from British publisher Boosey & Hawkes). It offers select scores of 20th century and contemporary composers like Thomas Ades, Louis Andriessen, Bela Bartok, Harrison Birtwistle, Pierre Boulez, Sergei Prokofiev, and Igor Stravinsky.
Harvard Online Resources for Music Scholars Digital Scores is a continually growing directory of high quality online repositories of online sheet music.
Music Treasures Consortium is a collaborative effort between the Library of Congress, Juilliard School, and others. Researchers can browse and view scans of important music manuscripts such as Beethoven's opera Fidelio or Mahler's Symphony no.1.
Sheet Music Consortium, a joint effort UCLA and other universities across the country, provides access to scans of public domain sheet music. Popular song sheet music makes up a large proportion of this database.
Web Library of Seventeenth Century Music is an open-access, peer-reviewed collection makes available works composed between 1600 and 1700 that are not commercially available in print, and no editions have been posted online. The WLSCM is a service offered by the Society for Seventeenth Century Music.
Josquin Research Project - Don't be fooled by the name. This resource has music by famous early composers like Josquin des Prez, as well as many of his well known, and lesser known contemporaries.
Early Music Sources is a helpful site to help one cut through the massive amount of early music resources on the web. Rather than having much of its own content, it acts as an excellent database and index for finding relevant resources quickly.