Detect a new Android notification

Actually, it is possible, I use it in my app.

For Android 4.2 and below:

You need to register an AccessibilityService and make sure the user enables the service.

Example for a service:

public class InstantMessenger extends AccessibilityService {

@Override
public void onAccessibilityEvent(AccessibilityEvent event) {
    if (event.getEventType() == AccessibilityEvent.TYPE_NOTIFICATION_STATE_CHANGED) {
        //Do something, eg getting packagename
        final String packagename = String.valueOf(event.getPackageName());  
}
}

@Override
protected void onServiceConnected() {
    if (isInit) {
        return;
    }
    AccessibilityServiceInfo info = new AccessibilityServiceInfo();
    info.eventTypes = AccessibilityEvent.TYPE_NOTIFICATION_STATE_CHANGED;
    info.feedbackType = AccessibilityServiceInfo.FEEDBACK_SPOKEN;
    setServiceInfo(info);
    isInit = true;
}

@Override
public void onInterrupt() {
    isInit = false;
}
}

Example for checking if your Service is activated

For Android 4.3 and above:

Use the Notification Listener API

The new Notification Listener API in Android 4.3 enables you to do this.

With this there is less need for the accessibility hack. It also allows you to dismiss notifications.