Here are some core Thunderbird extensions that I frequently use. Most of them (if not all) can be found at the Mozilla Thunderbird Add-ons page.
Contacts Add BCC Button
If you use the Contacts side bar when you are writing an email, this extension can be handy. On the side bar, there are Add to and Add CC buttons. This extension adds a Add BCC button to the side bar.
You need to set up html signature files first. But once you do, in the extension's Options, you can add them in. Then, when you are writing an email, you can chose which signature to use via the icon on the toolbar. Handy if you have a polite one to use when writing to your parents and a less-polite one for when you write to friends.
This extension is great for if you receive a lot of emails that contain a lot of attached pictures. Instead of scrolling down to see each picture, you can click on the slideshow button (bottom right of the email - beside the list of attached files) and a slideshow viewer will pop up. Expand it to full screen and then click through the pictures one by one. The viewer window includes options for saving individual pictures or all pictures at once to your hard drive.
Provides a timestamp you can place anywhere in your email. Fully customizeable. This is handy for if a server delays your emails, or if a server puts the wrong time on your email. Make sure you have your computer set at the right time and time zone to start with though!
Same extension can be used for Firefox or Thunderbird. Set it to Check for updates when Thunderbird starts, Install updates when they become available,and un-check all the other boxes in Options. With these settings, all your Thunderbird extensions will update themselves in the background and you won't have to update them manually.
How to install a Thunderbird Extension