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Julian Faraway, in his book on linear model with R, used half normal plot to find hat(a value in hat matrix) with too large value.

That's the only example where I saw half normal distribution is used. But other than that, I can't come up with real application of it though I guess it might be useful for approximating some positive numbers whose mean is near zero.

What are real applications of it?

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Because the square root of a $\chi^2(1)$ variable is half normal, most applications involving chi squared values with 1 degree of freedom can be interpreted as applications of a half-normal model. – whuber♦ Dec 27 '11 at 21:10
The half normal is used in the EGARCH specification for asymmetric volatility modelling. – user11978 Jun 14 '12 at 3:41

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In quality control, there is something called a moving-range statistic which is the absolute value of successive differences. The half-normal serves as the basis of the chart as discussed in the following article:

http://www.tandfonline.com/doi/abs/10.1080/08982119508904612#preview

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In Bayesian statistics the half normal,with a sufficiently large variation parameter, can be used as a noninformative prior distribution on the SD of a standard distribution. This is suggested in, for example:

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian analysis, 1, 515-534.

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It's true that you use it, but if you do it's typically NOT noninformative, since then you could also use a more typical/easier noninformative prior like gamma or the uniform. The half-t makes more sense as a weakly informative prior, which has a distinct tendency but wide enough tails to let itself be overwhelmed by the data in the face of strong evidence. – Erik Nov 13 '12 at 12:00

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