Mozilla is not only a great WebBrowser, it is also a great MailClient. Trust me on that one.
I have found MozillaMail to be a pathetically buggy MailClient especially when using multiple IMAP accounts on different servers. I would recommend PINE
I have found MozillaMail to be a perfectly stable MailClient, even when using multiple IMAP accounts on different servers. I have also been using it longer than 
MattBrown for this purpose. The only times I have had any issues with MozillaMail is when trying to force its folder hierarchy to show subfolders of INBOX to be at the same level. Personally I don't care about this, but MattBrown seems to think that a MailClient is only useful if it shows subfolders of INBOX at the same level as INBOX.
Note that this is arguably a bug in MozillaMail (not supporting the NAMESPACE extension to IMAP4 correctly), however in itself it is not a fatal bug. Some IMAP servers (recent versions of Cyrus for example) can do server-side namespace alteration, which MattBrown is used to.
I would like to point out to MattBrown that 'pathetically buggy' is fairly strong language to describe a MailClient that works 100% fine for other people, who have only had issues with it when trying to resolve a purely cosmetic issue.
While it is true that I have had issues with MozillaMail and CourierImap? displaying incorrect folder hierarchies my problems with MozillaMail have extended well beyond this. The following problems are some of the worst I have experienced with Mozilla and "standard" IMAP configurations. And no I don't run the latest beta versions.
MattBrownI've seen the second problem you describe, when I am using an IMAP server off a dynamic IP: eg my server at home. If it changes IP, mozilla sometimes requires a restart. If you are using SSL, mozilla always requires a restart. This isn't neccesarily a mozilla problem. It could perhaps handle it better, however dynamic IPs are not designed for running services off them. If you want a reliable mail server, get a static IP or host your mail on a machine with a static IP.
The former one does look like a bug, however I've seen issues like it relating to incorrect XUL. Try removing your .mozilla directory ( you did do this when you upgraded last, like the upgrade notes in Mozilla tell you to, right?) and reloading.
I used MozillaMail on my LAN at home and still had problems, so I don't believe that dynamic IP's are the problem.
I have removed my .mozilla directory countless times due to being unable to start mozilla, this only adds to my frustrations with MozillaMail.
If mozilla mail refuses to ask if you want your password remembered, make sure you don't have the line
user_pref("signon.rememberSignons", false);
in your mozilla prefs.js file!
And I really hate it how you can't highlight and select email addresses from within the header box.
