In past months, Samsung filed several patents for a smart wrist-watch with a flexible display. Samsung already confirmed they are working on such a device, but this is the first time we see an actual patent that may offer a glimpse into what this future device will look like. I don't know if the patents actually detail an OLED display - such a design could also use a flexible E Ink panel. Although I think that Samsung will probably opt for a full-color OLED.
Apple is also working on such a device - at least according to rumors. Back in December 2012 we reported on the iWatch: a bluetooth smart watch that has a 1.5" touch PMOLED, apparently made by RitDisplay. In February 2013 the New York Times and the Wall Street Journal both posted about this rumor. In May 2013 Taiwan's Economic Times reported that Apple had made some prototypes with a 1.8" PMOLED, but decided they were too large and they will indeed opt for a 1.5" display.
If all these stories are correct, Samsung will have a huge advantage over Apple's device with the flexible OLED. Of course the company will first have to start producing flexible YOUM OLED panels in volume...
Meanwhile, a new California based company called Emopulse embarked on a very ambitious project - to build a smart watch that uses two flexible OLED displays. The company hoped to raise $300,000 using crowd funding to develop this watch (the 128GB model will cost $480, you can see a concept above) - but I don't think they managed to raise the money as the page has been removed since.