After looking around the web for answers to this problem, none of them helped.
The solution I did find, however, seemed to solve my problem perfectly well.
My problem: hotmail/live e-mail accounts were unable to connect to pop3.live.com (i.e. you could see in Thunderbird status bar "Connected to pop3.live.com" but, nothing would actually happen).
To solve this problem, I took the following steps...
1. Right click on the e-mail address in the left-most addresses/accounts pane that's having issues.
2. Click "Settings"
3. Click on "Security" for the security sub-section of settings for this account
4. Click "View Certificates"
5. Click on the "Others" tab
6. You should see an avast! certificate there for the particular pop3 mailserver.
7. Delete it.
8. Close Thunderbird completely
9. Open avast!
10. Click on the "Security" tab
11. Click on the "Mail Shield" icon
12. Click "Settings" (next to Stop)
13. Go to "SSL Scanning" in the navigation menu
14. Uncheck "Scan SSL connections"
15. Click "OK"
16. Click "Settings" again (next to Stop)
17. Go to "SSL Scanning" in the navigation menu
18. Check "Scan SSL connections"
19. Click "OK"
20. You should get a warning you'll need to restart your e-mail client. If you haven't already closed it, restart it now.
21. When Thunderbird attempts to retrieve your e-mails, you should get warnings about certificates. Accept them all (make sure permanently). These are just avast! generated, you can check by clicking "View" and you will see the publisher/creator is avast! Mail Scanner.
22. Enjoy your e-mail (hopefully)
By deleting the certification, and what seems to force avast! to re-generate their SSL certificates, it should now be working fine and scanning your e-mails downloaded over SSL.
IMPORTANT: There is a lot of misinformation about removing ports from the Redirect Settings in the Troubleshooting section. Do not do this. If you have done this, replace them with the following:
HTTP port(s): 80,8080,8091,8081,8008,8888,3124,3127,3128
Check "Ignore local communication"
SMTP port(s): 25,587
POP port(s): 110
IMAP port(s): 143
NNTP port(s): 119
Check "Ignore local communication"
Hope this helps someone. If it does, please post and let others know.