Falsifiability is very important in science; scientific theories and hypotheses can not be scientific unless they are falsifiable. When creationist finally learn this they attempt to use it to discredit the theory of evolution.
Before I tear apart the argument I should explain what falsification is and its relevance in science. If something is unfalsifiable it can not be shown to be false, if it is falsifiable is can be shown to be false. Science does not prove theories and hypotheses, when scientists test theories and hypotheses, they test them in an attempt to falsify them. This is why falsification is so important in science.
A static fossil record
Rabbits in the Cambrian, Monmouths in the Jurassic, Dinosaurs in the Pleistocene, etc. However we don't see that in the fossil record, instead we see certain animal and plant fossils in certain layers that that were laid down at certain times.
The discovery of chimeras
While related species do share features they are not chimeras as they inherited those features from common ancestors. A true chimeras would be an animal that has combine features from multiple animals, features that could not have been inherited from common ancestors nor evolved though convergent evolution (some features will evolve multiple times (eyes, wings, flippers, limbs, etc)). For example: a mammal with feathers, a snake with fur, a bird with a mammalian reproductive system.
Some will point to the platypus as an example of a chimera how ever it is not. Its duck bill is only looks similar to a ducks bill, it is actually an enlarged snout covered with thick leathery, soft skin. Platypus do lay eggs but they are not the only mammals lay eggs; Platypus are part of order called Monotremes all of whom lay eggs.
Experimental data that shows no changes in populations under controlled conditions
Some organisms reproduce fast, very fast, fast enough to actual witness evolution of these organisms. Place populations of these organisms in controlled conditions and observe them over a long enough time span long enough for evolution to occur. These experiments have been done and they shown evolution in action.
A mechanism that prevents mutations or prevents mutations from being passed down
Mutations are a prime factor in evolution, they cause the modification that natural selection acts on. If mutations were impossible or couldn't be passed down evolution could not work, there would be no modification. The problem is: we have seen mutations, beneficial mutations and have seen them passed down.
Evolution can be falsified, it hasn't been because the theory is accurate.
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