Most people who use informal logic in debates make mistakes from time to time. That's perfectly normal and exactly what we would expect from human logicians; however no one misuses, misunderstands and/or misrepresents logic quite like a Christian apologist. Here's a list of some common 'misses' made by apologists and explanations for why they are fallacious or just plain wrong:

1) In the ontological argument for the existence of god there is an attempt to define god into existence. Most of us have heard that that simply doesn't work, but do we know why? There is a huge difference between an analytical claim or an axiomatic proposition and a synthetic claim. It is true that all bachelors are unmarried because we define bachelor to mean an unmarried man. And that's perfectly fine. If you define bachelor to mean a black cat, as long as we both agree on the definition, then you can say all black cats are bachelors. Norman is a black cat and therefore he is a bachelor. By definition. Put this is simply a peculiarity of language. And there are no synthetic propositions being made. In other words, nothing is being proposed about the external world. Is is simply axiomatic. If I said that all bachelors live in California that would be a synthetic proposition. In order to assign a truth value to a synthetic proposition we need to ask, what prediction can be made based on this proposition? If there is a bachelor living outside of California then the proposition would be said to be false. If there is nothing about the synthetic proposition that has any pragmatic predictability then it is incoherent and would be said to be false. If I define a boohblah to be a sentient pink crystal floating in intergalactic space that's fine, but if I then tack on: "that exists in reality" to the definition it has no bearing on whether or not it actually exists in reality. The first part of the definition is axiomatically true, any sentient pink crystal floating in intergalactic space would be, by definition, a boohblah. But the latter part would make it a synthetic proposition which needs to be demonstrated empirically otherwise it is, by default, false.

2) All incorrigible propositions are true.
I feel warm.
I hear music.
I see red.
I believe x.

They are simply internal thoughts, feelings and perceptions. They cannot be wrong. However this fact has been misused by countless apologists. To say that "I feel god in my heart" is essentially meaningless to the external reality.
To say that you have a feeling is an incorrigible proposition. You can even define heart to mean mind and god to mean love and then they would axiomatically be absolutely true. So what? Congratulations, you feel god in your heart. You cannot use that to prove your synthetic propositions. It's equivalent to saying: "I hear music" no-one can say you don't. But if you say I hear music because the neighbor is playing music on their stereo it becomes different. Now you can predict that if the neighbor is playing music on their stereo that if you go to their house you would see it on and hear it louder. If you do not than it's false. If you do, it's true. If you cannot... I think you know where I'm going with this.

3) Begging The Question. The Moral argument for the existence of god. To assume the truth of the conclusion in one of your premises in a syllogism is a logical fallacy known as begging the question.
"One: If god does not exist then objective moral values do not exist.
"Two: objective moral values do exist.
"Three: therefore god exists. "--William Lane Craig

You see what he did there? He's assuming the existence of objective moral values... Tying the existence of objective moral values to the existence of god and then concluding god.

Further:

If not x then not y. Y therefore x. Let's plug in some other words to this and see what we get: If humans don't exist then language does not exist. Language exists therefore humans exist.
That does not follow. And simply asserting that something is contingent upon something does not make it so. Back to the argument; there is another assumption being made: that objective moral values exist. How does he know this?
Can we make a pragmatic prediction based on this synthetic claim? Yes: if objective moral values do exist then none will ever disagree on what is moral and what is not. So if people disagree we would call the proposition false. Oh but Craig says it transcends beyond human opinion. Well, in that case the claim is incoherent and by default has no way of being determined true or false by humans and the null hypothesis becomes the rational stance to take.

I was going to display more logical fallacies and misuse of logical syllogisms by apologists but I realized that this is already quite lengthy so I will break it up into sections.

I look forward to both atheist and theist feedback on this subject and I'd be more than happy to explain anything that I have not given adequate explanation of so far.