Can I mute sound notifications, but allow chat heads on Facebook Messenger? On Facebook Messenger, on my Android phone, I want to permanently mute audio notifications for new text messages, but I would still like to receive a chat bubble.

Is this possible?
The trick is to look in the Android settings for this, rather than in the Facebook messenger. 

In Settings -> Notifications it will list all of your applications, and the notifications settings they have. 

Some applications will have several notifications tfor the given app. 

In the 'chat and calls' notifications - you can disallow sound. The downside here is that this will also prevent Messenger calls from making a sound.