HTML5 Notification not working in Mobile Chrome

Try the following:

navigator.serviceWorker.register('sw.js');
Notification.requestPermission(function(result) {
  if (result === 'granted') {
    navigator.serviceWorker.ready.then(function(registration) {
      registration.showNotification('Notification with ServiceWorker');
    });
  }
});

That is, use ServiceWorkerRegistration»showNotification() not new Notification().

That should work on Android both in Chrome and in Firefox — and on iOS in Safari, too.

(The sw.js file can just be a zero-byte file.)

One caveat is that you must run it from a secure origin (an https URL, not an http URL).

See https://developer.mozilla.org/docs/Web/API/ServiceWorkerRegistration/showNotification.