hi @katy, thanks for your note.
I have push notifications working (the code in main.ts is currently commented out) right now using Firebase with a NativeScript Angular app here: http://www.github.com/jlooper/yowwlr
I find that backgrounded notifications right now are buggy on iOS specifically, - on Android they seem to work fine. This is stopping me from writing a blogpost on it, though, but working on it.
There are other push solutions out there if you don't want to use Firebase. OneSignal is one, and we have a plugin for it on plugins.nativescript.org. I've used Pushbots with great results in the past too.
For your own use case, you could initialize the push service in main.ts, and then copy the push messages into a database to display permanently (until the user deletes them, anyway) in your app's inbox. Curious what you are building!