One of the best known sites for music scores in the public domain:
www.imslp.org-

Free offerings from one of the best classical music resources online.  Most songs available with MP3 files and in multiple formats (PDF, MIDI, Scorch, etc.) 
http://www.virtualsheetmusic.com/free.html

Pianist and master pedagogue Walter Cosand graciously shares his amazing public domain score collection online:
http://waltercosand.com/CosandScores/

Free public domain scores, available in an eclectic variety of file formats (PDF's, Finale, etc.):
www.sheetmusicfox.com

A resource where registered members "share" music scores with each other:
www.pianofiles.com

The Horowitz Scores online:
a free resource mainly devoted to transcriptions by the legendary pianist Vladimir Horowitz:
http://www.vkgfx.com/

Remarkable resource for Chopin scores:
http://www.vkgfx.com/scores/chopin/posthumous/

Free scores of works by J. S. Bach:
www.FreeSheetPianoMusic.com/bach

Sibley Music Library at the Eastman School of Music
https://urresearch.rochester.edu/handle/1802/291

Wikifonia is a place for musicians to publish and collaborate on lead sheets:
http://www.wikifonia.org/

Classic Ragtime Piano resource:
http://www.ragtimepiano.ca/

Free Piano Collection for Beginners:
http://www.take-a-piano-sheet-music-break.com/free-piano-collection-for-beginners.html

Free Piano music in graded levels:
http://gmajormusictheory.org/Freebies/freebies.html

American Sheet Music resource from the Library of Congress:
http://memory.loc.gov/ammem/mussmhtml/mussmhome.html

Mozart Resource: Neue Mozart-Ausgabe Online:
http://dme.mozarteum.at/DME/nma/start.php?l=2

The Henselt Library of Nineteenth-Century Piano Music at European-American University:
http://www.henseltlibrary.org/sources.html

Unusual collection of rare salon music:
http://www.mindspring.com/~lesmuskey/OLD/http://www.mindspring.com/~lesmuskey2/OLD2/

Mostly Music - resource for Jewish Sheet Music
http://www.mostlymusic.com/all_sheet_music.php

Classic Cat - free classical catalogue
http://www.classiccat.net/

The Classical Archives - scores available with paid subscription
http://www.classicalarchives.com/

Bach Cantatas Website
http://www.bach-cantatas.com/index.htm

Indiana University Sheet Music Collection
http://www.letrs.indiana.edu/cgi/b/bib/bib-idx?c=devincent;c=starr;g=sheetmusic;cc=ALLSELECTED;xc=1;
sid=162264b16c49133120e8f7eafa49a158;page=index

Chopin Early Editions Collection at the University of Chicago Library
http://chopin.lib.uchicago.edu/home.html

Complete Beethoven String Quartets transcribed by Louis Winkler for Piano 2 hands
http://filer.case.edu/~dss/

Bach Gesellschaft - site in English and Korean
http://www.greatjsbach.net/Work.php3

Online Opera, Song, Orchestra, Choral,and Chamber Music Scores, among others:
http://www.dlib.indiana.edu/variations/scores/

Wagner Operas
http://www.wagneroperas.com/indexwagneroperas.html

Links to a huge selection of sites offering contemporary/pop piano sheet music
http://my-piano.blogspot.com/

Scorelib is a search engine to easily find free sheet music.
http://www.free-scores.scorelib.com/

Other Free Music Score Websites:

http://www.greatscores.com/

http://www.pitt.edu/~deben/freebies.html

http://www.sheetmusicdigital.com/freepdffiles.asp

http://www.musicstudents.com/class.html

http://www.music-scores.com/piano/composer.php

http://www.allpianosheetmusic.com/free_piano_sheet_music.htm

http://www.sheetmusic1.com/new.great.music.html

http://www.easysheetmusic.com/popular_classics.asp

http://www.freesheetmusic.net/sheet_music_downloads.html

http://www.mfiles.co.uk/sheet-music.htm

http://www.mutopiaproject.org/browse.html

http://www.sheetmusicusa.com/piano/begin.htm

http://www.sheetmusicarchive.net/

http://www.musicnotes.com/free/

http://www.easybyte.org/