Generally when one looks at general overviews or outlines of the various arguments for and against God, the arguments are classified in a neat order. However, I have found that the classifications for theistic arguments (in particular) are often prone to counter-examples.
For example, it is often said that what makes ontological arguments what they are is the fact that they are a priori in nature. The problem with this claim is that some, like Graham Oppy, argue that there are experiential ontological arguments. In addition, there are also arguments that are a priori in nature that aren’t ontological arguments. For instance, Kant gave an argument-in the period before he was skeptical of pure/theoretical/speculative reason- that was a priori in nature but doesn’t exactly strike one as an ontological argument. The argument that Kant had in mind claimed that God accounts for the notion of possibility.
The cosmological argument is often classified as an argument that starts with experience and the senses. However, it’s not clear that this is true of all cosmological arguments. The argument from contingency is obviously a cosmological argument, and certain arguments from contingency are purely modal arguments; modal arguments are arguments that are a priori in nature (or at least they don’t have to be grounded in experience).
So, obviously the ontological, cosmological, and teleological arguments are “the big three” arguments when it comes to arguments for classical theism. However, in the past few centuries moral arguments have risen to the level of being ‘worthy’ of a fourth category. It’s true that these four categories don’t account for all theistic arguments, but they do cover all/most of the popular arguments you will often hear. By way of example, arguments from religious experience argue that people having religious experiences is some evidence for God’s existencem, but this isn’t an argument that falls under the four list above. (i.e. the existence of religious experience makes God’s existence more likely than it would have been had we not has cases of religious experiences). The former argument is just one example, and I think it is by far one of the more modest arguments from religious experience. 
Classifying Arguments Against God
It doesn’t seem there has been as much work done to classify non-theistic arguments (or arguments against God) as there has been for theistic arguments. Nevertheless, there has been some work done there.
Firstly, there are dystelelogical arguments. These are the evil-twins of telelogical arguments. And like telelogical arguments, they come in different flavors: fine-tuning, biological, scale, etc. I actually wrote a post about them here.
Secondly, there are arguments from nonbelief. These include arguments for atheism being the default position, including strong and weak versions of atheism. Also, a sub-category of arguments from nonbelief is arguments from divine hiddenness (an argument that I’ve written a lot about). Arguments from divine hiddenness don’t claim that God exists and is hiding; rather, they are concerned with the alleged conflict between some type of nonbelief and the concept of God as all-loving/all-good/all-just, etc. In addition, there are arguments from nonbelief that deal with the objective evidence available out in the world; the fact that the evidence for the God of classical theism is weak or ambiguous is, perhaps, a justification for believing that such a Being does not exist (i.e. if God did exist, there would be strong evidence or more evidence of her existence).
Thirdly, there are ontological disproofs of God’s existence. For instance, some of these arguments mirror classical ontological arguments. On the other hand, some of these proofs are conceptual or definitional disproofs of God. Personally, I don’t find most (if any) of these arguments to be too compelling, but they are interesting arguments. It makes sense to classify some non-theistic/atheistic disproofs as ontological, including arguments that are conceptual and definitional. But whether all conceptual or definitional disproofs are ontological disproofs might not be agreed on by all. Certainly I think there could be some a priori disproofs of God that aren’t ontological arguments. By way of example, modal arguments for atheism are a priori in nature, and it’s not plausible that all these are ontological arguments (e.g. modal arguments from evil).
Finally, the most apparent category is arguments from evil. Within arguments from evil, we have arguments that focus on human pain and suffering, but we also have arguments that just focus on animal pain or suffering that isn’t the result of human behavior like diseases; hurricanes; etc. In addition, there are social evils and political/economic evils. Furthermore, there’s arguments from evil that focus on the failure of theodicies and related arguments that focus on the lack of God’s felt presence during horrific suffering.
 Nevertheless, I think the argument cherry-picks the facts about what we know about religious experiences, specifically the distribution of religious experiences.