The idea that the Earth is a flat plane seems to have become slightly fashionable in conspiracy-theory circles. As skeptic Michael Marshall has pointed out, proponents of the idea, also seem to espouse well known tropes around the illuminati, Biblical creationism, Biblical numerology and big-government controls. Historically it was never really a 'thing'. Most people in history - and now - would have lived their lives without really giving much thought to the shape of our planet.
The different designs of a flat-earth put forward by modern proponents vary quite substantially but the commonest idea is that the earth is circular with the north pole in the middle and a ring of ice around the edge or south pole. Of course, this fails to explain the myriad of facts that would be wrong if this was true - solar movements, time differences, triangulation distances - for instance.
It is easy to be dismissive of proponents as cranks and fringe conspiracy theorists but as an example of where critical thinking can lead if taken to uber-extremes, allied with the power of social media and YouTube it is a classic one.