The most common fallacy I see is personal incredulity, in other words they are too damned lazy to do the work even though they always claim they did the work. Flat earth and creationist nuts are the most likely to be this dishonest.
First, odds are not calculated based on "this seems highly unlikely," they are calculated based on samples, the larger the sample size the more accurate the odds are. For example, evolution has a 1 in 1 odds of happening, every organism we encounter varies from it's parent organisms.
Creationists used a failed formula that only works for humans, and only if evolution did not happen. The value they cite is only if the DNA started from scratch each time, so no evolving, no parental traits, no biogenesis.
The result is 1 in 600+ billion, which is correct if the DNA was built from nothing by manually randomizing the order in which each chemical was introduced. In other words, the creationist cites odds of abiogenesis producing a specific species spontaneously.
They are lying, plain and simple, a strawman that isn't even their own work. Asking them where they come up with the number always results in more lies such as "I did the math myself."
Flat earth nuts are even worse about this, they will make stupid assertions like "gravity wouldn't work on a globe." Asking them to show the math results in them spamming you with nonsensical videos and strawmen.
The fact is, gravity is omnidirectional, thus clusters of rocks in space will become somewhat spherical. The only reason Terra is not a perfect sphere is because of plate tectonics, which is the result of our molten core.
Plate tectonics has actually kept the planet looking like a giant asteroid, if you remove the water from the surface. So no, Terra is not a perfect sphere, but it is spherical, or a globe to be more precise.
The discovery of Terra being spherical came long before we launched into space, one of the earliest recordings was in ancient Greece. Plato was one of the earliest to observe phenomena that could only happen on a curved surface.
It was eventually proven using shadows, by placing pillars at great distances on the same altitude, they measured the shadows at different times in the day. The only explanation for their findings was a spherical Terra.
That is what people in science call "showing your work," or more simply put "empirical evidence." These simple tests can be duplicated and they will always show the same results, that part of the process is what religious people fear most, in science it's called "peer review."