Wilcoxon signed-rank test or One Sample t-test? I conducted a training session and for each of my users, I measured pre and post score (before and after training session). Each of my users (n=28) is measured twice, so my data is paired:
Data:
user pre-score post-score
1    10        30
....

Hypothesis:
$$
H_0: \mu_{Post-Pre} \leq 0\\
H_1: \mu_{Post-Pre} > 0
$$
In order to check the hypothesis, whether the post-score is significantly higher than the pre-score (whether the training had a positive effect on users' score), I would conduct a paired t-test (t.test(Post, Pre, mu=0, conf.level=0.95, alt="greater") in R). 
Problem: The post data is not normally distributed. That's why I would go with a non-parametric test, i.e. Wilcoxon signed-rank, instead of the paired t-test.
However, if a calculate the difference between the pairs (Post-Pre), I end up with one sample which is normally distributed. So I would have to test the following hypothesis in a one sample t-test (t.test(Post-Pre, mu=0, conf.level=0.95, alt="greater") in R):
$$
H_0: \mu_{Diff.} \leq 0\\
H_1: \mu_{Diff.} > 0\\
$$
Question: Which method is appropriate? Should I use the non-parametric paired Wilcoxon text because my post data is not normally distributed? Or can I work with the difference of the pairs (one sample) which is normally distributed and use a one sample t-test?
 A: First, it's not necessary that the differences be Normally distributed, just that the sample mean of the differences be (pretty close to) Normally distributed.  The Central Limit Theorem lets us know that the distribution of the sample mean converges to Normal as the sample size gets larger under a broad set of conditions, which your data certainly satisfy.  For example, calculating the sample mean of a Uniform distribution with 12 observations was, in the very old days, one of the ways used to generate approximately Normally distributed random variables.
Second, the Wilcoxon signed-rank test doesn't test the same thing as the paired t-test.  It tests whether the differences follow a symmetric distribution around 0.  Thus, substantial asymmetry can cause the test to reject the null hypothesis even if the true mean is zero.  (In your case, given that your analysis of the differences indicates Normality is plausible, this isn't likely to be an issue.)  
In your case, I suspect the two tests will give the same result, but I'd go with the t-test.  With 28 observations of data that isn't very far from Normally distributed, you should be fine.
