Which icon sizes are required for progressive web apps (PWA) as of Q1 2018?

If you want to include a complete set of icons for Android:

icon-72x72
icon-96x96
icon-128x128
icon-144x144
icon-152x152
icon-192x192
icon-384x384
icon-512x512

There is some helpful tools like https://app-manifest.firebaseapp.com/, to create icons.

For iOS, you will need:

icon-120x120
icon-180x180

With square background (can't be transparent background). A good repository for references at https://github.com/gokulkrishh/awesome-meta-and-manifest

As per Google Developers

icons must include a 192px and a 512px sized icons

Here is our blog on the same.