For most of the twentieth century, belief in a flat earth was in serious decline due to a lack of evidence. Today, thanks to the disruptive and often ridiculous power of the Internet, there is an amusing increase, with three common types of adherents. Those who engage with the idea as a bit of a harmless joke, those who engage in classic trolling pretending to be flat-earthers whilst secretly testing the knowledge of their interlocuter, and those who literally believe in it. Surprisingly, most "flerfers" are in the latter category, and come from a perspective of selective Biblical literalism mixed with conspiracy theories.
Whilst the evidence for an oblate spheroid Earth (quite close to a sphere, a bit squashed on the poles and bulging at the equator) is quite overwhelming, the physical problem with flat earthers invariably comes down to their astounding inability to comprehend scale. Often, they will argue, for example, that airline pilots would have to dip the nose of the plane to account for the planet's curvature, and that isn't observed. Well, matter of fact, pilots do exactly that - about 22m for every 1000m travelled. The Earth, it turns out, is quite big relative to the size of the human observer.
Assuming that a flat-earther does actually accept that space exists (many don't), their adherence to flatness does actually have a cosmological equivalent; the galaxy is, in fact, relatively flat. Now, this might sound like "flat galaxy theory", and indeed it is, it is grounded in evidence. To the best of our knowledge, our galaxy started as a cloud and, with a modest spin, conservation of angular momentum causes it to spin faster, leading to a collapse to a flattened disk. Further, the solar system's relative flatness is formed in a similar manner (Mercury and Pluto have a bit of a tilt, but mainly it's disc-like, with the planets having elliptical orbits).
So if the galaxy and the solar system are flat, why aren't planets? Well, for a planet, gravity pulls in all directions once it is large enough, forming a spherical shape. Internal gravity dominates rather than orbital motion. Gravitational equilibrium causes a planet to become spherical, whereas for a solar system or galaxy, the flatness occurs because of the motion and distribution of orbital material shaped by angular momentum. "Distribution of orbital material" is pretty important; you can stand on a planet, but it's a bit harder to stand on the solar system or galaxy - there are some significant gaps between the material - as the classic "moon as a pixel" website illustrates.
Here's another interesting fact: everything is a bit of both. Even Earth is "a little flat" (around the poles) due to momentum, and Saturn is the "flattest" planet in the solar system. The star Altair is massive enough to be a spheroid, spinning fast enough to be flattened. It has a rotation period of 9 hours and is 22% wider at the equator compared to the two poles. Further, whilst galaxies don't stop spinning, this momentum can slow and change under certain conditions. For example, the galaxy could collide with another, and the angular momentum is redistributed, quite chaotically, especially if the two galaxies are of a similar size.
So instead of advocating for a flat earth, perhaps our flerfers should adopt the more scientifically sound "flat galaxy theory" and just think a bit bigger. And, for the truly brave, they might want to look into the current observations of the cosmic microwave background, which suggests that the universe as a whole appears to have something close to a zero curvature. Flat solar system, flat galaxy, flat universe. But the Earth, like all other planets and suns, remains a sphere (more or less).