I think it was Anovis that asked for what are atheist-style arguments. It was a while ago. Sorry this took so long, I had planned to respond but I didn't want to rush it.
Atheists tend to spend most of their time being agnostics, disproving the arguments of theists of why God must exist ("
Design requires a designer", "
Law requires a Lawmaker", "
Complexity of life occuring by accident is highly improbable"). These deserve a separate response of their own, though you'll probably see them come up in most religious discussions.
However, there are some occassions when atheists can go on the offensive and state that God
cannot exist (this is sometimes called 'strong atheism'). Agnostics should note that this is different from proving a negative ("There is evidence that God does not exist!"), which is impossible; it is dealing with logical contradictions in the common definition of 'God', of why the existance of God is impossible, before we can even begin to consider physical proof (existence of God is a priori impossible).
There's a few ways strong atheists do this.
1. God is all-powerful
"
If God is all-powerful, can he create an object that is so heavy that he cannot lift it?
If yes, then God cannot lift it and is not all-powerful.
If no, then God cannot create it and is not all-powerful.
Therefore, God cannot be all-powerful."
This is a nice one to start with. It isn't really an argument for why God cannot
exist, rather than argument for why you cannot be lazy with definitions and why the supposedly awe-inspiring quality of omnipotence is actually meaningless. Responses to the self-contradictory argument are usually things like "God can do anything logic allows" or "God is outside of logic" (this response is quite false, but for a rather long and elaborate reason). Either way, it is making us reconsider what 'God' actually is, and that is the purpose of this argument.
2. Problem of Evil
"
If God is all-good, all-knowing and all-powerful, why does evil exist?
If God knows about evil, but does not have the power to undo it, God is not all-powerful.
If God knows about evil, but does not want to undo it, God is not all-good.
If God wants to undo evil, but does not know about it, God is not all-knowing.
Therefore, God cannot be all-good, all-knowing and all-powerful, given the existance of evil"
There's lots and lots of different ways this one is given, the Epicurean paradox is the nicest phrasing (he was disproving the existance of multiple gods). They all deal with reconciling the existence of evil or suffering with the existence of a God that should be willing and able to end evil and suffering. I've tried to structure the argument in the least elegant and most systematic way (and we're also ignoring the self-contradictory argument above, or rather including it into the argument). This one is covered in theology with respect to what 'evil' is, and why evil cannot be an absolute quality in itself. (Instead 'evil' might be a 'lack of good', but that doesn't really address the argument much either).
3. Argument from Divine Hiddenness
"
God exists and:
1. wants all humans to believe he exists before they die;
2. can bring about a situation where all humans believe in him before they die;
3. does not want anything which would conflict with and be as important as his desire for all humans to believe he exists before they die;
4. always acts in accordance with his most important desires.
If God were to exist, all humans would believe so before they die (from 1).
Not all humans believe God exists before they die.
Therefore, God does not exist.
"
This seems like an elaborate one, but it is similar to the Problem of Evil in that it deals with the inconsistency between the real world and the world as it would be if God had certain desires, as well as the power to see them through.
As with the self-contradictory argument, the only way to conclude that God exists is to abandon the premise. In this case, the premises aren't just dealing with the properties of God, but with the properties of the real world as well, so they could be the rejected premises. Quite a common response is to reject the sentence "
Not all humans believe God exists before they die", and suggest that atheists don't really exist or that everyone naturally believes in God. As an atheist, I think this is absurd, and a little distressing as I'm not entirely sure how to prove that I actually am an atheist, but as they cannot prove that I secretly believe in God (without touching on any of the other arguments), that rejection isn't going to go anywhere.
There's lots of discussion that can be spawned from this one in particular, but none of the premise rejections are logically coherent (they spawn other discussions, relying on other assumptions. Quite often this assumption is that God exists, when the very argument is to determine whether he does.)
4. Argument from Free Will
"
If God has free will, then at any point in time he may either choose to do a certain thing or choose not to do it.
If God is omniscient, God knows everything that will happen in the future, including all of the choices he will make at any future point in time.
If God knows what choice he will make in the future, he will not be able to make the opposite choice.
Therefore, God cannot be both omniscient and have free will.
"
There's another version regarding the free will of humans, but the logic is the same: if God knows what we are to choose before we choose it, how are we free to choose otherwise?
This is one that can't be solved simply by redefinition, in that it requires you to redefine 'free will', 'knowledge' and 'choice', and as these aren't concepts reserved merely for theology (like 'omnipotence' is), theology cannot redefine them.
A common response is that God does not exist within the restraints of linear time, but this doesn't resolve his 'free will', as omniscience requires that God has knowledge of all things in all times, even if his perception of time is radically different to ours.
5. The Transcendental Argument for the Non-existence of God
Largely a response to TAG. I can't really do it justice in few words, read up on it here
http://www.infidels.org/library/modern/michael_martin/martin-frame/tang.html and other places. It largely deals with wooly concepts of 'morality' and the like. I don't really like it.
6. Chicken and egg
"
If all things must be created, then God must too."
This one results either in an infinite series of Gods creating Gods, or the rejection of the premise and the conclusion that not all things require a creator. If God doesn't require a creator, why does the Universe?
x. Invisible Pink Unicorns
"
Invisible Pink Unicorns are beings of great spiritual power. We know this because they are capable of being invisible and pink at the same time. Like all religions, the Faith of the Invisible Pink Unicorns is based upon both logic and faith. We have faith that they are pink; we logically know that they are invisible because we can't see them." -- Steve Eley
IPU is a satiric parody religion, revolving around the notion that the Goddess takes the form of a unicorn that is paradoxically both invisible and pink. It is a parody of religion and the supernatural in general. Debates among followers are absurd and tortuous, satirizing many religions all at the same time. The overall 'point' of IPU is that there is no more reason to believe in a Christian God than there is in the Invisible Pink Unicorn.
Reductio ad absurdum with heavy emphasis on 'absurdum'.
These are a few of the nice little arguments. There are others that generally deal with physical evidence, and are about 'bad design' and that if everything around us really was created, it wouldn't be quite as messy and disorganised and generally unbeautiful. These ones are excellent for discussion, as they can touch upon biology, quantum mechanics, string theory, mathematics, the whole lot; but they're not really very elegant arguments. And then there's the Anthropic Principle, which is used as
both evidence for and against the existence of God. That one deserves a thread to itselt.
If you only accept one or two of those strong atheist arguments, it will still result in a radically different 'God'. If God is not all-powerful, or all-good, or all-knowing, why is he God?
A radically different 'God' is exactly what strong atheism doesn't cover (yet), and polytheism in particular can open a whole new field of possibilities. Of course, all the gods still have to avoid self-contradiction (so none of them can be all-powerful), but they can avoid the Problem of Evil if one of them were all-good (but not all-knowing) and another was all-knowing (but not all-good). This alone would result in a fascinating mythology, and might create whole new logical problems.
This is the sort of thing that makes strong atheism interesting; following nothing but logic can take you down wholly unexpected paths.