The only argument against theism that is worth considering at all seriously as an argument against theism[note 1] is the inductive problem of evil.
The inductive argument from evil holds that it is unlikely that God would have created a world containing the degree, kind and amount of evil that we observe.
Now, one might offer the inductive argument from evil as an argument for some supernaturalistic hypothesis, such as that there is an evil god (either alone or along with a good or neutral one), or that there is an indifferent god, or the like. In that case, what I will say below does not apply. But, among contemporary philosophers, the primary hypothesis competing with theism is not one of these unorthodox supernaturalistic ones, but naturalism. And then the philosopher offering the inductive argument from evil needs to show not just that the degree, kind and amount of evil that exists in the world is unlikely on the theistic hypothesis, but that it is more unlikely on the theistic hypothesis than on the naturalistic hypothesis.
But to show this is tricky. Much of this point comes from C. Stephen Layman (I am grateful to Todd Buras for letting me know of Layman's argument). Let us grant for the sake of argument that the conditional probability of the degree, kind and amount of evil that we observe is, say, 10−20 given theism. Denote theism by T and let E is the event of there being the degree, kind and amount of evil that we in fact observe. Then it's granted for the sake of the argument that P(E|T)=10−20. Do we know that the probability of the degree, kind and amount of evil that we observe is more than 10−20 given naturalism, i.e., that P(E|N)>10−20, where N is naturalism?
Many of the evils that we observe are of such a kind that they are only evil because of conscious life—pains are like that. Therefore, E entails the existence of conscious life, but says a lot more than that conscious life exists. Hence: P(E|N)<P(C|N), where C is the claim that there is conscious life. If P(C|N) is no greater than 10−20 then we do not have P(E|N)>10−20. Thus the naturalist to use the argument from evil against theism must be able to estimate P(C|N) as greater than 10−20. I do not think we are in a position to make that estimate. And without it, we are not in a position to know that the inductive argument from evil supports naturalism over theism.
Moreover, many of the evils that we observe are moral evils. That there are moral evils entails that there are morally responsible persons, and that there are moral truths. IF R is the claim that there are morally responsible persons and M the claim that there are moral truths, then in order to use the inductive argument from evil as an argument against theism and for naturalism, one would have to show that P(R and M|N) is bigger than 10−20. But I don't think we are in a position to know that. First, it may be very unlikely that persons would arise given N. Second, it seems quite plausible that for moral responsibility one requires persons who engage in a form of causation very different from the natural causation around them—if all we have is the kind of causation that ordinarily goes on in nature, we don't have moral responsibility. If so, then the likelihood of R given N is nil, or perhaps very small. Furthermore, these persons would have to have moral beliefs to be morally responsible. But to secure intentionality for moral beliefs is a tough task for the naturalist. It's hard enough to secure intentionality for empirical beliefs on naturalistic theories, but perhaps some kind of causal theory of content can be made to work. But it does not seem likely that in a naturalistic setting (except an Aristotelian one—and that's not what people usually mean by "naturalism") we could get intentionality for moral beliefs.
And while I do not know of an entirely convincing argument that the existence of moral truths is incompatible with naturalism, the existence of moral truths is not as prima facie likely given naturalism as given non-naturalistic hypotheses like theism. (In fact, theism entails the existence of moral truths.)
Moreover, among the evils that E reports, there are some truly horrendous moral evils. While bare-bones theism may have difficulties with explaining these, naturalism likely has a difficulty with explaining them, too. But theism, unlike naturalism, is at least compatible with positing a supernatural tempter, a devil, to help explain these horrendous moral evils. So, once again, it is not clear that naturalism does better.
Of course one can tell stories about the evolution of persons, responsibility and even the capacity for horrendous moral evil. But are these stories any better than stories that the theist can tell—stories about divine design, the value of human and demonic freedom, etc.?
If I am right, then the argument from evil does not support naturalism. The naturalist can still offer the argument as an ad hominem, but if she is serious about thinking that the argument is a good reason to reject theism, she needs to be open to the possibility that it is a good reason to abandon naturalism.