I want to sort and filter my email into folders based on rules that I define based on "from", "to", the email account that received the message, maybe bayesian spam rules, and so on. Kind of like what Outlook Express does for you when you use POP3. Except that I don't use POP3. So I need something that will work for me anyway, either an IMAP Windows client that will move messages around for me, or something that lives on the server.
It's a plus if it can actually let me make rules by which account of mine actually got the email, which is a rule that seems to be lacking in Outlook Express. (I have several forwarding email addresses forwarding to the same one actual address, and I'd like to be able to tell them apart in my filtering, even if I got an email via BCC.)
Thunderbird may be able to do this, but I'm also interested in non-Thunderbird products that can do this.
Ease of use is another plus, but I'm not willing to sacrifice much filtering power for it.
For server-side programs, it's also a plus if the program is rather well-known, in case I decide to install it somewhere where I don't have root access, and I end up having to try to convince a sysadmin to do it for me.