While there is nothing like the beauty of a full-size printed manuscript, the wide-variety of free sheet music is available on the is an education in itself.
Indexes | Classical | Popular | Mood/EZ ListenChristmas/Folk | 2nd Tier
Free web sheet music:  A primer
Free sheet music of amazing quality and variety is abundant on the web, but getting it can be can be frustrating.  Manuscripts are in a variety of formats, often requiring "plug in's" to dislay and print the music; and there is no telling what you'll find before you get there.  But a little patience and persistence can put you in touch with some gems, especially if you are in a browsing state of mind. 
Index Sites:  These sites don't have music; they list free music sites.  Check them regularly for updates. 
Classical Content:  Classical music is easy to find (so much is in the public domain).  These are the best of this class.
  • Sheet Music Archive  A number of classical public domain scans, many with fingerings.  Good first place to look.
  • The Chopin Files  Most of what he wrote in good quality print, some fingerings, midi support, interesting background information. An A+ site.
  • Music Scores.com Good selection of composers, several difficulty levels, even some arrangements (e.g. flute & piano), full midi support.
  • Werner Icking Music Archive (formerly GMD Music Archive)  Not primarily piano, but a variety of serious classical manuscripts for keyboard, strings, vocal; some midi support.
  • Daly Music = My Sheet Music Numerous classics (other tunes, too); for purists, authenticity of classic scores might be a question mark.
Popular:  For free popular music, enjoy what you can get; it is unlikely you will find a particular number or arrangement you might have in mind.
  • OddRealm  More pop tunes; when you click a number your are taken to the site where the number was originally found.  Numerous dead links, though.
  • Music of Yesterday  Straight forward voice with piano accompaniment of 200+ popular classics (in Noteworthy Composer format to look and listen). 
  • PDMusic  If your definition of popular extends to what might have been sung at home just after WWI, check these scans of "popular" songs.
Romantic, EZ Listen, New Age: 
Folk & Christmas:
Other fine sites, especially at the price, but not my first tier:
  • Sheet Music Online A little awkward to use, but some goodies for piano and strings tucked away.
  • Mel Bay has a monthly freebie for piano.
  • Virtual Sheet Music  Has a few free samples (classical mostly); great quality print
  • Mutopia  Small but growing collection of short classics with midi support.
  • Masters of Classical Music  Limited selection, but some lesser know items from Bach's Notebook for Anna Magdalena
  • Any Score.com  Small collection of samples, most can be found elswhere
  • Musica Viva   Grainy print with poor layout; good early keyboard focus.
  • Sheet Music USA  Grainy print with poor layout; variety plus midis.
  • 8Notes.com  A lot of guitar mixed in, and demonstrates riffs more than complete pieces.
  • Composer.com  Most content duplicated elsewhere; worth a look.
  • Piano Public Domain  Might find these classics elsewhere; maybe not.
  • Nissimo  Some teaching pieces plus you can submit your own work.