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Means that something is very fine, or that you feel excellent. If someone asks how you feel and you simply say "fine", you might just be saying it out of convention. Adding the "dog's hair split three ways" shows that you actually mean it.

Another form of the expression is "finer than a frog's hair split three ways", although literal frogs do not have hair.
You: How are you doing today?

Me: Why, I'm doing finer than a dog's hair split three ways!
by JCEG November 25, 2006