IMO, the only strong argument against the Christian God is the problem of evil. But isn't it a fallacious argument from incredulity?
1. Given the nature of the world, I cannot imagine how a good and all-powerful God could possibly be true;
2. Therefore, there is no good and all-powerful God.
Firstly:
The problem of evil is an internal critique that seeks to show a contradiction between certain tenets of a given God, religion, or so. For instance:
Religion A says such-and-such and the Gof of religion A is X, Y and Z.
Such and such, and X, Y, Z entail contradictions because of this-and-that.
Therefore, something is amiss about religion A/religion A's God.
The PoE does not, and must not look towards "the nature of the world" in and of itself. It has to deal with statements by a given religion about "the nature of the world."
Secondly:
You can only have an argument from incredulity if certain meaningful propositions are being denied because, and simply because, of incredulity. Actually showing that certain statements do entail contradictions based on, say, logic, is not the same as denying something simply because of incredulity.
And lastly:
This is not to say that the PoE is a successful or an unsuccessful internal critique of course, but just that, if done properly, the PoE would be an internal critique.