On mobile apps, where/When is the best time to popup this "Rate Us" dialog? I know that it may change from game to game, but is there any statistics out there?
It would all depend on the type of app that it is and the type of user that is using the app. The best time to remind the users to rate your app is when they have finished some task. This is the time where they will feel positive about it and they would be willing to rate it. i.e. If it's a game, display the rating when they have finish a level that is a milestone. If it's a tracking app, i.e. running app, you can count the frequency and set some rules. i.e. a person runs >X times a week, the app will ask once a month at X point in the app.
I can't give you precise numbers because it really depends on the use of the app. If the user opens the app everyday be careful with showing the pop up too much because users will get annoyed.
What is also good to do is that if the app has asked XX times, never ask again. Let's face it. Users will not rate it in the future if they have been asked 20 times in the past.