Stephen Law is perhaps the foremost defender of the argument from suffering. While not an argument for atheism per se, if the argument is successful, then it does constitute a sound a defeater of classical theism, which holds that God is omnipotent, omniscient, and perfectly good.
One aspect that Law focuses on is the suffering of animals. Animals, many believe, are innocent of any moral crimes, and so it would be unjust to allow them to suffer. More precisely, it would be unjust to allow them to suffer to the extent that they do. Since the God of classical theism would not allow such suffering (according to the argument), and this suffering is real, it follows that the God of classical theism does not exist.
Now, one classical theistic response to this is to say that while animals suffer pain, they are not aware of the fact that they are in pain. If true, this minimizes the emotional impact of this aspect of the argument from suffering, and severely undercuts its rational import. After all, suffering is either just or unjust only if there is some level of self-awareness, or at the very least a potential of self-awareness.
Law, like other defenders of the argument from suffering, responds that (at least some) animals do have the capacity for this kind of self-awareness. Instead of contesting this response, I want to accept it for the sake of argument. Why does this result in an odd consequence?
The problem is that with self-awareness comes a recognition of moral obligations, or (again) at least a potential for this recognition. If animals have self-awareness, then it is reasonable to think they are cognizant that certain things are right and others wrong. This in turn means that animals are morally culpable. Yet, we find among animals many moral atrocities, such as rape and murder. Ordinarily we would refer to an ape intentionally causing the death of another of its kind "killing," as opposed to "murder." However, moral culpability changes all of that.
Unwittingly, then, Law and other proponents of the argument from suffering have actually given classical theists a further justification for animal suffering. For "all have sinned and fall short of the glory of God" (Rom. 3:23) and "the wages of sin is death" (Rom. 6:23). If animals have self-awareness and they are cognizant of moral obligations, then it follows that they are capable of sin, which (like for humans) is their metaphysical cause of death.