Android - Lollipop - mute notifications but not calls

It can't be done. https://code.google.com/p/android/issues/detail?id=79445

This is currently a very hot topic with many people extremely frustrated that this functionality has been removed. Hopefully google will bring it back.

Here's how to do it:

1. Press a volume button to get a menu with three items, "none", "priority" and "all".
2. Select "priority"
3. In the dropdown that appears, tap the cog for Interruptions settings (you can also access this through Settings)
4. Under "priority interruptions", turn on "Calls" and turn off the other two options.

That should work, but I haven't tested it.

EDIT: Apologies, I missed the part about your still wanting vibrations and LED notifications from things other than calls. I suspect there isn't a way to do that built in. It may be possible using something like the Light Flow app.

This is now possible using stock Android using do not disturb mode. You can set do not disturb on, but still allow calls and alarms. Works on a Pixel 2 at least.

You won't get vibrations from messages, but the status led will still light up.