While there has been a lot of new development on Firefox for Mobile aka Fennec there is currently no official Thunderbird for Mobile. Further, there has not been any talk about developing a mobile version of Thunderbird. This could change in the future as the development of Thunderbird is now under Mozilla Labs, a group that specializes in these types of ‘projects’.
The closest you can get to Syncing is an add-on, Zindus which allows you to sync your GMail and Thunderbird contacts. There is no synching of mail or calendar. Two options here, using a web-based email/calendar service like GMail or use an email provider that allows IMAP (of coursed this syncs messages only, not calendars or contacts).
I should add it is also possible to use a POP email account but you will need to make sure in your local Thunderbird under the Server Setting section Leave messages on server is checked. This will allow you to retrieve your messages both on your desktop Thunderbird and mobile email client. Downside to this though if you delete the message locally, you will also need to so on your mobile device and vice-versa. Further your folder settings won’t be synced either.
Will this get better in the future? Unfortunately, I really doubt it as the growing trend with email is moving away from desktop email clients more towards web-based access. The only exception is Exchange Service which syncs everything (contacts, messages and calendar). While many mobile devices do support Exchanged email, it is a Microsoft technology and is not supported by Thunderbird (you would need to use MS Entourage or Outlook).