I've recently been reintroduced to the Flat Earth Society, and while intending to study for an ethics exam ended up sitting with a friend for roughly an hour coming up with different reasons why the earth must be round.
To begin, let me quickly paraphrase a few of the key points that the Flat Earth Society holds:
1. The earth is actually a large disk with center at the North Pole, and surrounded by an insurmountable and heavily guarded (by our own governments) ice wall.
2. The earth does not produce any gravity, in contrast to all other heavenly bodies which do. To compensate, the Earth and the universe maintain a constant acceleration of 9.8 m/s^2. On Earth this translates to a net downward force of 1 G on all its inhabitants.
The following is a list of complications that arise from a flat Earth model. It is by no means exhaustive and some of them are refuted by the Flat Earth Society in any number of different ways. They are in particular, reasons that I did not see a solution to here. I invite anyone to add more or dispute any one of them.
1. The existence of the magnetic North and South are not explained by a flat Earth model. Moreover, the gradual change in the exact location of these poles is not explained by a flat Earth model.
2. Ocean currents, particularly the Antarctic Circumpolar Current which circulates Antarctica, does not fit in to a flat Earth model since the period of revolution of this current would have to be many times larger. (Since it would have to circulate roughly the circumference of the flat Earth).
3. Given that the Earth is accelerating upwards, the gravitational equivalent must be constant at all points on Earth regardless of their altitudes. Thus the force experienced on Everest would be equal that felt at the Dead Sea and any point on the surface of the Earth.
4. Since populations have increased dramatically since the Earth's inception, and since mass and energy are, to some degree, interchangeable, and finally, since the Earth receives most (if not all) of its energy from the sun, not all of which being reflected back into space, it is only reasonable to suggest that the Earth is increasing in mass. Thus considering that F = ma, then the Earth cannot continue to exert a force of 1 G on us without increasing its acceleration. This would be impossible since it would violate the law of conservation of energy.
So from this it should be pretty obvious that the Earth is actually a large torus centered around an enormous microwave oven. See, I can do it too.
Sources:
http://en.wikipedia.org/wiki/Antarctic_Circumpolar_Current
http://en.wikipedia.org/wiki/Flat_earth_society
http://www.alaska.net/~clund/e_djublonskopf/Flatearthsociety.htm
http://theflatearthsociety.org/forum//