You have to open the settings menu -> tools -> "sign message".
Afterwards you can choose an address which you want to use signing the message and enter your message.
Sometimes it shows an error message. In this case you should try to update your firmware and try it again afterwards.
But I have a Nano, not a Nano S.
Excuse me for my typo. I meaned the Nano. I own a ledger nano myself (nano, not the new nano s).
And signing message works as i wrote above.
I would not recommend to import your private keys into bitcoin core. You would have to use your recovery seed to create the
private keys to be able to import them to bitcoin core since ledger does not support key exports.
Additionally this would completely destroy the sense of having a hardware wallet which is not directly connected to the internet.