Hello Alan,

My replies in Situ below in colour:
Before attempting any of my suggestions below, I would suggest you backup first.

> On 26 May 2019, at 12:24 pm, Alan Smith <sma...@iinet.net.au> wrote:
> 
> All personal mailboxes in the Favourites bar are greyed out on a new iMac (27 
> inch 5K with Mojave 10.14.5).  I Used Migration Assistant to copy all data 
> from Time Machine. The backup has been erased.

When you ‘Add a mailbox’ to the The ‘Favourites Bar’ by dragging the mailbox 
from the Mail Sidebar, Mail doesn’t actually move the mailbox; it adds an alias 
for it to the Favourites bar.
You could have lost your link to the server.
Check the Mailboxes you dragged to the Favourites Bar from the Sidebar are NOT 
Disabled?
Go to Mail > Preferences > Accounts tab - check they are all online.

If all accounts are online & you still have the Favourites mailboxes greyed out.
Drag the greyed out mailboxes out of the bar and they will disappear, then drag 
new ones from the sidebar to replace them.

> 
> I Rebuilt the inbox for my main mail account but this made no difference.   
> ~/Library/Mail/ contains V5 and V6 folders. 

The V5 folder is used by High Sierra.
Before doing anything, make sure you have a current backup.
1. Quit Mail and move the V5 folder to the trash, but don't delete it. 
2. Open Mail and verify everything still works OK and all your messages are 
there. 
Once confident everything is OK, you can empty the trash.

> 
> I normally move mail from my Inbox to the Favourites mailboxes.  I have mail 
> archives (outside the Mail app) up to December 2017.  The Mail “Sent” folder 
> holds all (?) messages from 2017. 
> 
> Can I recover my Favourites mail history?
> 
> Regards
> Alan

Cheers,
Ronni

13-inch MacBook Air (April 2014)
1.7GHz Dual-Core Intel Core i7, Turbo Boost to 3.3GHz
8GB 1600MHz LPDDR3 SDRAM
512GB PCIe-based Flash Storage

macOS High Sierra 10.13.6

> -- The WA Macintosh User Group Mailing List --
> Archives - <http://www.wamug.org.au/mailinglist/archives.shtml>
> Guidelines - <http://www.wamug.org.au/mailinglist/guidelines.shtml>
> Settings & Unsubscribe - 
> <http://lists.wamug.org.au/listinfo/wamug.org.au-wamug>



-- The WA Macintosh User Group Mailing List --
Archives - <http://www.wamug.org.au/mailinglist/archives.shtml>
Guidelines - <http://www.wamug.org.au/mailinglist/guidelines.shtml>
Settings & Unsubscribe - <http://lists.wamug.org.au/listinfo/wamug.org.au-wamug>