Plasmas are still nowhere as good as a CRT for contrast as they don't have near as dark blacks. The dark areas still contain a high amount of high frequency noise. They also are a linear device using PWM, so they require electronic gamma correction which can compromise dark area detail. Even with later generation technology, they are prone to burning. Their green phosphors tend to be bluish which lowers color accuracy, but this can be corrected with color management.
LCDs have come a long way to catch up with plasma and exceeds it in some respects. Burning is not a problem, but retention can be. This seems to vary between displays from non-existent to almost as bad as plasmas. However they will tend to fade out after a while though I have seen some that never quite do. Colorimetry of LCDs have reached to extended gamuts. Ghosting has nearly been eliminated though it's still a limitation for alternating image display of 3D. Viewing angle has vastly improved but blacks still become raised when viewing off angle. For casual viewing it's not that big of a problem.
DLP and LCOS are limited to projection. While the quality is very good, the projection requirement limits how thin the display can be made for rear projection. Consumer rear projection screens tend to be plastic which creates color sparkling on bright scenes. LED illumination improves DLP as a mechanical wheel is not required, and improves both technologies with expanded color gamut. DLPs use PWM, so using a single device in sequential color mode can compromise the amount of gray scale levels. Like plasma it depends on the shortest duration possible in the PWM.
OLEDs have great contrast comparable to CRTs, have great colorimetry, no burn or retention issues, no viewing angle issues, no dark area noise. The main issue has been the deterioration of the individual colors. On cell phones this isn't as much of an issue as they tend to have a limited lifetime. But on a computer or video display, especially at the high price premium, this could reduce their appeal. However this has improved tremendously. They are still a linear display so PWM is required, but these devices can have a very short duration. OLEDs look to possibly be the winner of the direct view display race, especially with their thin displays.
If color e-paper can match the speed and quality of other display technologies, it may become the ultimate winner.