Windows Phone: Icons not loading, Internet Explorer just black, performance poor

My Windows Phone the other day started acting badly—any icon from an application I installed refused to load, Internet Explorer would just show a black screen, and the performance was VERY poor. The cause? I’d used this awesome device too much and had run out of space:

oops

Using the Zune software was the only way to see that!

What had happened was that I hadn’t installed new apps, but I had done a bunch of updates the day it started, and so it seems the updates filled it up. Once I cleaned up some unused apps & games, I freed up a few gigs—and everything started working fantastically again!