If you ever encounter a flat Earth believer in real life, and especially if they happen to have a flat Earth map at hand, here are some arguments and questions you can present to them using that map, and also an additional bonus argument at the end. (You could of course also present these online, but I think these are much more effective in person, because you can explicitly show and demonstrate these to them using the map.)
This is one of the maps most commonly used by flat-earthers (it's directly from the Flat Earth Society), and the arguments are based on it:
The size of Australia vs. the United States
Most flat-earthers, especially in the United States, no matter how uninformed and illiterate, know of their own country as well as Australia. And these two countries are conveniently almost the same width.
The width of the United States is 2680 miles (about 4313 km). The width of Australia is 2511 miles (about 4042 km). They are pretty close to each other in width, with Australia being just a tad bit narrower. Indeed, if you compare their sizes from an actual map, they are pretty close:
We can measure the width of both countries in real life using several methods, and I don't think that even they would doubt that the method of simply driving from one and to the other, measuring the traveled distance, is a good one.
On their map, however, and even disregarding how squeezed the country is, Australia is significantly wider than the United States:
They might respond that the map is just an approximation, and in reality Australia should be narrower on that map. However, this would cause a problem with longitudes. (Longitudes are those straight lines radiating from the pole towards the edges.)
The east coast of Australia is at a certain longitude and the west coast at another. These longitudes can be measured. Again, there are several ways to do this, known for hundreds of years, but one of the easiest ones is to measure with synchronized clocks exactly when it's mid-day (ie. the Sun is at its peak altitude) on both coasts, and what the time difference is between these two events. Knowing that the Sun makes one whole revolution around the Earth in 24 hours allows us to know pretty exactly the longitude coordinates of both coasts.
The same can be done, of course, in the United States. The difference in longitudes gives us another measurement for the width of both countries, and we find out that they are very close to each other. Even just the time difference between mid-day on both coasts does this.
The longitudinal lines on that flat earth map are pretty accurate, as seen for example from this:
Both countries span slightly over three longitudinal lines.
According to the flat earth map, and according to the longitudinal lines, Australia should be significantly wider than the United States. However, according to their measured widths (which can be measured physically) Australia should be a tad bit narrower than the United States.
Equinox argument 1
The equinox is the time of the year when the Sun is directly above the equator (which happens twice a year). This is a particular situation in that both day and night are almost precisely 12 hours each everywhere on Earth. In other words, the Sun illuminates pretty much exactly half of the Earth in terms of the longitudinal lines.
This, of course, can be easily measured. No matter where you are on Earth, both day and night are almost exactly 12 hours each during the equinox, which means that the Sun is visible and not visible about the equal amount of time.
On the flat earth map this would mean that the Earth is illuminated by the Sun like this:
Since flat-earthers believe that the Sun is some kind of spotlight some tens or hundreds of miles above the Earth's surface, ask how exactly it's able to illuminate the Earth like this. How is it possible that a source of light hovering above a planar disc like this, would cause that kind of shadow. Note how the light extends much farther on the left and right sides of the image than on the center. Why would light from the Sun travel over twice the distance, eg. in this example to reach the southern parts of South America, than it does to reach the North Pole? Why would it stop at such a clear and sharp line, cutting at the pole?
Equinox argument 2
Also during the equinox, no matter where you are on Earth, the Sun will always raise directly and exactly from the East. Again, this can be easily measured. The direction of North, East, and where the Sun raises from, can be easily, accurately, and pretty mundanely measured.
On the flat Earth, however, this poses a big problem. This is illustrated below:
The arrows start from points on Earth, where the Sun is seen raising. The green arrows show the direction from which the Sun is seen raising from, ie. directly from the East. On the flat Earth model the Sun should, however, look to raise from the direction of the red arrows.
The difference is quite extreme. For example, at the equator, at Central Africa, the Sun is observed to be raising directly from the East (green arrow), but according to the model it should be observed to be raising directly from the North-East (red arrow). A deviation of 45 degrees.
In South Africa the deviation should be even larger than 45 degrees.
This model of the flat Earth could be easily corroborated or disproven by going to the equator, or South Africa, or Australia, or pretty much anywhere in the southern hemisphere, during the equinox, and seeing where exactly the Sun raises from. If it raises directly from the East, rather than North-East, then this flat Earth model cannot be correct.
Washington DC, in the United States, and Melbourne in Australia, are approximately at the same latitude, but on opposite sides of the equator. This means that during the respective summer solstice at both places (ie. in June for Washington DC and December in Melbourne) the length of day, from sunrise to sunset, is almost 15 hours. How is this possible on the flat Earth model?
Not only would the Sun need to traverse at a faster velocity in December than in June, but it would also need to illuminate a significantly larger area during December than during June. During the summer solstice in the southern hemisphere (ie. in December) the Sun would need to cover 15 hours out of 24 of the latitudinal line (ie. the big circle that goes around the Earth) that passes through Melbourne, ie. 62.5% of it.
Even moreover, in December the North Pole is in eternal night. The Sun never raises during the entire day. In other words, the Sun would need to illuminate 62.5% of the latitude passing through Melbourne, while not illuminating the North Pole at all. Thus, the illuminated area would have a kidney-shape, something along the lines of this:
How exactly does the "spotlight" Sun illuminate the Earth disc in this manner?
Traveling to Antarctica
Flat-earthers have this conspiracy theory that if anybody tries to travel to Antarctica, they will be stopped by some secret military fleet consisting of thousands of ships, and ordered to turn around at gunpoint.
This is all kinds of a hilarious conspiracy theory, but on this day and age it would be quite easy for them to prove right or wrong: Simply organize a trip to Antarctica, using their own ships and their own people, and livestream it the whole way. If they are indeed stopped by some secret organization military vessel, it would be caught on livestream before they could stop it.
It's actually funny how little effort these people are willing to spend in order to try to corroborate their wild claims. (I suppose that deep inside they don't want to actually try to corroborate it, for the fear of being wrong and thus ashamed, even though they would never admit this out loud.)