Answer to "Backlight randomly turning off"27" Imac Mid 2010 running El Capitan. Started having the same issue when I updated to Sierra last year. Reverting back to El Capitan solved the issue until this morning, where it started flickering before going completely black. The addition of "SMC Fan Control", and operating the fan speed at higher RPM's brought the screen back. Brightness must be 100%, otherwise it still flickers. It definitely seems like a heat dissipation issue, all things considered.
Answer to "Random Black screen, Help please"I recently ran into the same problem with my iMac (27-inch, Mid 2010) 2.8 GHz Intel Core i5 ATI Radeon HD 5750 1024 MB computer w/ 16 GB of ram. The screen started flickering than blacking out all together shortly after updating from El Capitan to Sierra. I tried Resetting Vram, Pram, using a different keyboard, booting in safe mode, etc. Nothing worked. In the end resetting the computer back to El Capitan was the solution. No issues experienced since reverting! They do not offer hardware support for this iMac any longer, so I was glad to get back to having a monitor that functioned normally. A couple of years ago I had the power supply replaced, but have had few other issues with it since. Hope to get a few more years out of this computer, as it is more than powerful enough for the work I do here at the office. Best of luck to those experiencing the same issue.