Every time I try to start kmail it starts but immediately stops and appears this popup message "Program stopped. Error: could not create collection trash resourceid: 3". Before for months all was working OK and I don't remember to have done some change in configuration. Somebody could give me any advice?

You may want to fire up "akonadiconsole" and check, under "local folders" (right click and select Configure Natively) if they point to a valid path.

I have done but I receive the same message.
The first time I tried to start akonadi appeared this popup message: "Failed to connect to database:Can't connect to local MySQL server through socket '/tmp/akonadi-guido.E4pUd3/mysql.socket' (2) QMYSQL:Unable to connect"
I restarted akonadi and it worked fine and I did as suggested.

I had the same problem. I configured the default Trash to be for a remote (IMAP) folder, and then when KMail tried to start I got

The Email program encountered a fatal error and will terminate now.
The error was:
Could not create collection trash resourceId: 4

This makes it impossible to use KMail, or even to reconfigure KMail from KMail.

This is similar to
viewtopic.php?f=215&t=120777&p=308452&hilit=KMail+trash+fatal+error#p308452

If you read past the rather desperate advice to run an SQL query, then the solution is to restart akonadi, but it is not explained how that is done.

I solved the problem by rebooting.

Akonadi can be restarted with "akonadictl restart" or from "Server > Restart" in akonadiconsole.