The Sonos app does not ‘play” music, so there is no audio generated to send via Bluetooth to the AirPods. The Sonos app tells the computers on the speakers where to go to get the signal. 

The only headphones that can be connected to the Sonos app are the Sonos Ace headphones - and then only to connect it to a soundbar or check battery status. This will not work. For all intents and purposes the Sonos app is a remote control only.

Thanks. What a shame. It seems like an obvious and common capability of most systems these days. 

I don’t get it either. They have a music service, but no way to use it with headphones, even the Ace. 

As has been said the Sonos app is just a remote not a music service. Any music played through your Sonos system is via content stored on your device or streaming service playing to your device. Sonos just provides a way to enjoy the audio over Sonos speakers as a single or grouped through your home. 

AirPods are for private listening which is through a BT capable device or connected Apple product such HomePods, AppleTV, iPhone, iPad or Mac. Simply put it’s a peer-to-peer hardware connection.

The only way would be to make the Sonos system broadcast its audio to BT, so airpods (or any BT headphones) could catch it and play it.

And the only way to do that would be to use a BT transmitter connected to Sonos Port/Amp. Something like Sonos App → (tells a Sonos sytem what to play) → Sonos Port/Amp → (converts Sonos audio into analog audio) → BT transmitter → (broadcast the Sonos analog audio into a BT stream) → airpods.

In other words, a BT receiver such as AirFly might work? 

https://www.twelvesouth.com/products/airfly

Yup, any BT transmitter (not receiver) will do. Then you just need to pair your airpods with your BT transmitter (something similar to https://discussions.apple.com/thread/7800150?sortBy=rank)

It a transmitter, not a receiver… but yes, that device would work IN COMBINATION with Sonos Port (or the older Connect (on S2 just devices manufactured 2015 or later). 
A Sonos Amp WON‘T work because of missing line out.