What sort of evidence is there that the earth is or is not flat?

At this point, there is such overwhelming evidence that the earth is round and revolves around the sun that the only way in which it makes sense to discuss flat earth theories is in historical context. 
Before the advent of modern technology and science, or understanding of gravity, many cultures considered the earth to be flat. Much of this was simply intuitive, due to the fact that it appears flat as we walk or ride horses or sail across short distances in sail boats. Although many Mesopotamian and other early cultures considered the world flat, as explorers traveled across larger distances and measured the positions of the sun from different latitudes, they began to realize that the earth was in fact spherical, but sufficiently large that its curvature was not readily observable by the naked eye. By 240 BC, the Greek scientist Eratosthenes had calculated the radius of the earth with a fair degree of accuracy, and increasingly accurate calculations of earth's radius and its distance from the sun became common in antiquity and have continued to be improved upon until the present.
The first circumnavigation of the globe was completed by Portuguese explorer Ferdinand Magellan in 1522 by ship. With modern technology, circumnavigation of the earth has become common, as have direct observations of a spherical earth from space. 
https://en.wikipedia.org/wiki/List_of_circumnavigations