What are the popular methods for outlier detection in univariate data, which do not assume normal distribution?
Generally, you should avoid trimming outliers in an ad hoc fashion and instead use nonparametric or robust alternatives. A recent review with Monte Carlo studies can be found in Bakker and Wicherts (2014). At least in psychology journals, Z-score cut-offs were most popular. Of course, I wouldn't recommend that; the simulation studies in the same article demonstrate that Z-score cut-offs can inflate Type I error rates.
Although the review is focused on independent samples t-tests, most of their recommendations will apply more broadly. They concluded with the following recommendations:
• Correct or delete erroneous values.
• Based on prior research, it is not recommended to use Z scores to identify outliers. We recommend methods that suffer less from masking like the IQR or the MAD-median rule instead.
• Decide on outlier handling before seeing the results of the main analyses, and if possible, preregister the study at, for example, the Open Science Framework (http://openscienceframework.org/).
• If preregistration is not possible, report the outcomes both with and without outliers or on the basis of alternative methods.
• Report transparently about how outliers were handled.
• Do not carelessly remove outliers as this increases the probability of finding a false positive, especially when using a threshold value of Z lower than 3 or when the data are skewed.
• Use methods that are less influenced by outliers like nonparametric or robust methods such as the Mann-Whitney-Wilcoxon test and the Yuen-Welch test, or researchers may choose to conduct bootstrapping (all without removing outliers).
Bakker, M., & Wicherts, J. M. (2014). Outlier removal, sum scores, and the inflation of the type I error rate in independent samples t tests: The power of alternatives and recommendations. Psychological Methods, 19(3), 409-427.