We all must admit despite the latest iPhone's using LCD panels, these are the best looking displays so far on the market and it proves that Apple's "tweaking" of "normal hardware" has it's rewards as I have the iPhone 7 plus and the screen is the best I have seen from Apple. Yes Samsung have been using AMOLED (Active Matrix Organic Light Emitting Diod) and OLED for many years, but their screens and imagery are too over-saturated to a point it looks artificial and the human eye can pick this looks fake straight away as it's not natural - too overemphasised.
The info read about OLED so far indicates plastic is used for curved displays, and glass is used for flat displays, as naturally plastic is more malleable / flexible. Glass can be bent using heat however this would be a little heavier and slightly more expensive. As for it breaking, they could make it tempered as well (heat treated) like safety glass on cars. Getting technical: Samsung also use Pentile Matrix Display OLED where the subpixels share the RGB colours, whereas RGB OLED use individual RGB subpixels - no sharing - which gives brighter colours and better contrast especially off-angle. Read the full details here:
http://tinyurl.com/hws6glf - this also explains why some panels have a bias towards blue / green and may appear off-colour when viewing from the side.
However, the drawbacks of OLED ironically are battery power. See LCD has a back-light which uses most of the power for the display, however for most uses its power usage does not drastically shift from black backgrounds to white backgrounds. While OLED has no back-light, it will consume around 40% of the power of LCD when displaying an image primarily black, and for the majority of images it will consume 60%-80% of the power of LCD. However an OLED can use three times as much power to display an image with a white background, such as a document or website.
Colour Balance Issues - OLED - Additionally, as the OLED material used to produce blue light degrades significantly more rapidly than the materials that produce other colours, blue light output will decrease relative to the other colours of light. This variation in the differential colour output will change the colour balance of the display and is much more noticeable than a decrease in overall luminance. This can be avoided partially by adjusting colour balance, but this may require advanced control circuits and interaction with the user, which is unacceptable for users. More commonly, though, manufacturers optimise the size of the R, G and B sub-pixels to reduce the current density through the sub-pixel in order to equalise lifetime at full luminance. For example, a blue sub-pixel may be 100% larger than the green sub-pixel. The red sub-pixel may be 10% smaller than the green.