Signing messages can be used to prove ownership of a specific bitcoin address.
A common situation where this might be handy is: You have your bitcoin exchange account protected with 2FA, but you lose access to your 2FA tool (mobile phone, email). To get your account unlocked, you can be asked by the exchange operator to sign some message with that bitcoin address associated with your exchange account.
In order to sign a message, select the Account in Trezor Wallet and press Sign & Verify in the top right corner. You should see a page like this:
1. Signing the message
Type in the message text into the Message field, then copy & paste the particular bitcoin addresses into the Address field (note it has to be a used address) and confirm by clicking on Sign.
2. Confirming the signature on your Trezor
Check the text of the message on your Trezor's display. If the text is correct, press the right button on your Trezor to 'Confirm'.
Once you have confirmed the signature on your Trezor, the signature will appear on the screen:
In order to verify a message, select the Account at Trezor Wallet and press Sign & Verify in the top right corner. You should see a page like this:
1. Verifying a message
Copy & paste the signed message into the Message field, then paste the bitcoin address the message has been signed with into the Address field, and finally, paste the signature into the Signature field. Click on 'Verify'.
2. Validating on Trezor
Trezor should show the beginning of the message on its display if verified correctly. Otherwise, you will get an error message on the Trezor Wallet page.
Once you have confirmed the verification on your Trezor, "Message verified" will appear on your computer screen: