According to Ethical Atheist
[…] medieval discoverers did NOT fear sailing off the edge of the earth. This thinking likely ended centuries before the age of Christopher Columbus […] the original works promoting flat earth belief that we did find are primarily pre-medieval. […] a flat earth view was not the mainstream view among scholars at all, but probably had a very limited following in pre-medieval times with only a trace in the earliest medieval times represented almost solely by Cosmas Indicopleustes. […]
It is clear to us that flat earth belief was extremely limited and mostly pre-medieval. The spherical view prevailed and is abundantly represented in works in later medieval times.
Update: According to Gregory Paul
Richard Thaler seems to think that the concept of a flat earth was widely held for a long time. This is not really correct. Mariners have long understood that the earth is strongly curved and possibly a sphere. Ships disappear down over the horizon (I once saw this effect on the Chesapeake bay and was shocked how fast the hull of a giant container ship dropped out of sight while the top of the superstructure was still easily visible). Polaris gets lower on the horizon as one sails south and eventually disappears and so on. Over 2000 years ago the circumferance of the planet was pretty accurately calculated by Eratosthenes using some clever geometry and sun angle measurements. This knowledge may have been lost in the west in the dark ages, but was well known to the Euroelites after the improved communications from Constantinople, Alexandria etc after the Crusades.
When Columbus was trying to get a government to cough up the money for his trip west he was not trying convince patrons that the planet was a sphere. The problem was that the experts told the people with the money that the distance from Europe and Asia across the super ocean separating them was 14,000 miles with no visible means of logistical support during the voyage (the perfect Bible did not mention extra continents being in the way). However, some works had come out saying that Eratosthenes had messed up and the planet was much smaller (I’ve heard this was based on Biblical passages and Columbus was very devout, but am not sure about that). Columbus figured it was 3-4000 miles to the west, a skip and a hop compared to the horrendous around Africa route. When the Spanish monarchs finally kicked the last Muslims out of Iberia and were having fun picking on Jews they decided what the heck and see what this Columbus fellow could do, the cost was just three little cargo vessels and their crews.
The story about the crews getting upset about sailing off the edge of the earth is probably a myth since they knew better. That Columbus was fighting the false knowledge of the flat earth apparently was invented in the late 1800s in an effort to make him a great American symbol of the progress of science over superstition associated with the 1892 celebrations.