Some of you may like to note these links:

Matty's 'Works of the Bard'

The actual first website to carry the complete works. You can access the plays only scene by scene, but the site does have a search engine which is a bit hit-and-miss, but usually dependable.

The MIT Complete Works

This claims to be the first website to carry the lot. No search engine last time I checked, but searchable on a play-by-play basis via Find >>> Edit. (Click on the play you want, then select "view entire play as single page.") You can also go straight to any scene, as at Matty's site.

Mr William Shakespeare

Loads of interesting stuff, links to info about the various editions, etc, etc. Absorbing site.