Update: I just found out that it is possible to check on the Supercharger status via the Tesla App. I had looked in what I thought were the obvious places (Charging and Charge Stats) but it's actually under Location. You will see what Tesla thinks is the current location of your car. At the bottom of the screen there's a tab you can swipe up named "Nearby Chargers". You will get a list of Superchargers in the area, addresses, maximum power at each location, the number of stalls available/installed and the distance to the stations. I don't see any of the 10 locations on my phone that indicates if there's any stalls out of commission but at least you can tell if the site is nearly full.
edit: Note that the App seems to only give you Supercharger status in the local area (<15 miles?) or the 10 closest. If you zoom out on the map you will not get status of stations outside that area. So if you want to know the status of a site 100 miles away that you plan on using, you'll have to use the car's navigation system.