If you buy a new iMac and are wondering if it might have one of the common problems here are some things you can use to check things over.
Dead Pixels / Yellow Screen
This flash program will run several tests on your screen. To use it click the Test Screen button.
The Check Pixels option will cycle through different color options on your LCD screen by activating all the red, green, or blue pixels separately and then displaying a white and black screen. If any pixels are stuck you can isolate them using this method. Apple considers a certain number of subpixel anomalies to be acceptable however (up to 20 or so?).
The Yellow Check will display two grey bars on your screen. Both bars should look identical in color. More information is available on the yellow tinge screen page.
The flickering screen issue involves random flickering or complete screen blackouts. People have suggested it may be caused by the temperature of the computer, a weak or variable power supply, having AirPort turned on, or the brightness of the screen (making the screen dimmer reduces the flickering). There appears to be no definitive explanation however (and as such no good way to test for this except to use your iMac a lot).
Apple has released a firmware update for ATI Radeon HD 4670 and 4850 graphics cards to address issues that may cause image corruption or the display to flicker.
Stuck iSight Pixels
To see if your iSight camera has any stuck pixels open Photo Booth and take a picture with a black cloth or piece of paper over the camera lens. In the resulting snapshot any dead pixels should be apparent as white dots.
Scratched CDs or DVDs
Insert a blank CD (or a disc you don't care about) into the side loading DVD player slot. If you push slightly towards the back of the computer as it loads you will likely end up with long horizontal scratches. To minimize the damage try to pull the disc towards you as it loads.
Offset Screen Display
The whole screen gets "shifted" down (offset) with the bottom part of the screen wrapping onto the top of the screen.
Production Information on your iMac
To get some further information about your Mac (such as specs, production info) you can use Chipmunk: Serial Number lookup (all you need to do is enter your full serial number). If you submit any of your problem information to them they will be able to track if there is a revision change by Apple and possibly alert you if your computer was part of a batch with other problematic computers.