Microsoft's Office 365 online office suite has an Outlook webmail client which was flawless well when I gave it a shot using Firefox. I think it costs $6/mo, no ads, no spying, because you pay for it. I don't know how well the search and spam filters work.
http://www.office365.com
Here's a demo video: http://www.youtube.com/watch?v=hOpCmUXLmTA
Note: This is _not_ the same thing as Hotmail.