iOS: Earlier this year, a little trick was discovered on iOS where if you attempted to download a large movie, iOS would clear up junk storage for you to make space for it. This no longer works in iOS 10, but I noticed last night that a similar thing happens when you download a large app.
If you're running low on storage, there's a good chance it's because you have some apps that are wasting a ton of space with their cache. In my case, I had several apps wasting about 1GB of space each, which is extremely problematic on a 16GB iPhone. When you attempt to download a large application, iOS will do anything it can to free up space for it, included clearing out the useless data in those apps and restoring them to their smaller sizes.
So, when you're out of space, attempt to download a big app. The app has to be bigger than the amount of storage you have free. Hearthstone is a good one to try because it's free and 1.89GB, but any large app will work. As you download it, head to Settings > General > Storage & iCloud Usage > Available. Here, you'll see your storage space go up and down, and eventually the size of each application should start shrinking. Go ahead and cancel the Hearthstone download after the space is freed up or delete it when it's finished downloading. Doing this, I went from 700MB available to 2.29GB.
This will work best when you're low on storage because of a few fat apps. If you're using iCloud Photo Library and have "Optimise Storage" turned on, downloading a big app will remove photos from your device first, then clean out storage in apps second. In a our tests, this worked best when the device was really low on storage, like my 700MB, but its seems to do the job as long as you're attempting to download an app bigger than the storage space you have available.