Remove file extensions in URLs with mod_rewrite but preserve 404 errors.

.htaccess

# The following will allow you to use URLs such as the following:
#
#   example.com/anything
#   example.com/anything/
#
# Which will actually serve files such as the following:
#
#   example.com/anything.html
#   example.com/anything.php
#
# But *only if they exist*, otherwise it will report the usual 404 error.

Options +FollowSymLinks
RewriteEngine On

# Remove trailing slashes.
# e.g. example.com/foo/ will redirect to example.com/foo
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.+)/$ /$1 [R=permanent,QSA]

# Redirect to HTML if it exists.
# e.g. example.com/foo will display the contents of example.com/foo.html
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}.html -f
RewriteRule ^(.+)$ $1.html [L,QSA]

# Redirect to PHP if it exists.
# e.g. example.com/foo will display the contents of example.com/foo.php
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}.php -f
RewriteRule ^(.+)$ $1.php [L,QSA]

I found CharlesHenry's comment helpful, but it only worked it I changed the placement of the directory name.

From:
RewriteRule ^subdirectoryname/(.+)/$ /$1 [R=permanent,QSA]
To:
RewriteRule ^(.+)/$ /subdirectoryname/$1 [R=permanent,QSA]