Ubuntu: LightDM Black Screen When Using SSD Workaround

Some Ubuntu users are reporting LightDM errors when using a SSD: about half the time when starting the computer, LightDM (the default Ubuntu display manager) doesn't load correctly and only a black screen and a blinking command line cursor shows up. The issue isn't limited to those who are using a SSD, but it's a lot more frequent when booting Ubuntu from a solid-state drive.
I too am experiencing this issue with Ubuntu 12.10 (not tested on older Ubuntu versions because I didn't have a SSD when I was using 12.04) and I've found two possible workarounds which have worked for me so I thought I'd share them with you, in case you're experiencing the same issue.