.forwardfile to send all my mail anywhere, or run it through a filter like procmail. With procmail, you can do almost anything. You can configure it so that when you send an email with the subject "reboot webserver", procmail will run a script that shuts down and restarts your webserver. It is an awesomely powerful and useful thing.
Then there's Gmail. I have nothing bad to say about Gmail. You cannot use it to run abstract programs — I've been playing with the idea of using Google Calendar to set up a remotely-configurable cron/at/batch equivalent, but it isn't there yet — but you have great stability, great interface and filtering. At this point. I'm going to state my position: If you can't filter mail, mail is useless. You'll get squashed by the deluge.
Which brings us to non-Gmail webmail. Specifically, my employer's webmail. You have to pay to get IMAP access to Yahoo Mail, which is a dealbreaker for me. My employer's webmail had fine POP and IMAP access, but what you couldn't do is filter on the server. You could configure clients to filter, but that means you have to copy all your filters to each client, whether it's the web interface, Thunderbird on this machine, Thunderbird on that machine, and so on.
My solution was to write my own filters to run on my workstation via crontab. It worked, and it allowed me to develop a means to specifically alert on things I find crucial rather than Thunderbird's "Alert if new mail" approach. But we've moved to a Zimbra-based solution which seems to have everything I'd want.