My Fiancée has developed the art of avoiding the point to a sublime level. I have learnt that it is totally fruitless trying to have a discussion about anything that isn't "happy-shiny".
She could argue for 24 hours straight without ever allowing the original point to be addressed. She is the master at steering away from the subject. It's very impressive.

She has me beat.
Having spent many frustrating hours being the recipient of these techniques, I thought I'd put some of them down on paper - with examples, either real or representative. This is a work in progress, so may change as I remember more. How many of these do you recognise. In yourself, or your SO?
1) Take your point and then make an extreme and preposterous example of it to shift the focus to the new version. This has two benefits: Firstly, it is much easier to argue against the extreme version, second, it avoids having to address the original point.
Example:
"I wish you wouldn't arrange nights out on a Friday night when we're supposed to be travelling to the weekend house. Can't you make it on another day?"
"You want me to stop seeing my friends. You never want me to go anywhere. You're not happy unless I'm in your sight."
2) Ignore the actual point of the sentence and latch on to a tangential point, or word in there. Argue that point. Again, shifting the focus away from the original point.
Example:
"Why did you lie to me about that thing the other day? It wasn't very loyal of you."
"Oh, loyal like a dog? I'll lick your hand when I come home. Loyalty is a word you use with dogs. I'm not your dog. Your dog doesn't mind if you don't feed it. She had no food in the bowl this morning."
3) Bring outsiders into it as unverified backup. Again, a double whammy here because you're now wondering how it is that she's discussing us with outsiders, yet refuses to discuss it with you. Takes you off at a nice tangent.
Example:
"I think most people would have been upset by what you said the other day."
"No, it's you. Most people wouldn't care. Jenny agrees as well.
4) Rarely let you finish a sentence. Butt in as soon as you start talking and inject some preferably random argument/complaint.
This has the double advantage that she doesn't have to listen to what you're saying and the randomness of the butt-in causes you to forget what you were trying to say in the first place. This technique is VERY effective.
Example:
"I think I should be allowed to...[Butt in here]"
"I asked you to phone the people about the roof. You didn't do it. How can I rely on you?"
5) Shift the focus of your concerns away from feelings to words and actons.
Example:
"I would like to feel that you do value our time together. You never say. If I ask, you won't tell me."
"OK. Give me the script of what you want me to say, and I'll read it back to you."
6 Make a fuss about the way the question/point is worded. Argue about the construction of the question/statement and thereby avoid answering it.
Example:
"That's a bit harsh, isn't it?"
"You've just answered your own question. What's the point in asking a question and answering it yourself?"
7 Answer a modified version of the question you ask. Outline the modified question, and then answer it.
Example:
"I had a great time out with you today. Did you enjoy it?"
"I enjoyed not being at work for a change, yes."
These techniques (and others that I'm probably forgetting right now) make it utterly impossible to have your say. I recognise them as they're being practised. I even point them out, but, of course, a judicial application of any of the above techniques makes it easy to avoid that point too.