What I know is that there are several non-parametric methods for trend analysis including the Mann Kendall test (Mann 1945; Kendall 1975), Spearman’s rho test (Spearman 1904, Lehmann 1975, Sneyers 1990), and Sen's Innovative Trend method (Sen 2012). Recently, there was also a new trend test (Onyutha 2021) published via http://dx.doi.org/10.2166/nh.2020.111 and also there exist a tool (CSD-VAT) to apply this recent method. There are also parametric methods, though because of their restrictive assumptions they are not that commonly applied.
Talking about the most commonly applied test (or the Mann-Kendall test), you can find so many of its variants including Hamed (2008), Hamed (2009), Yue and Wang (2004),Noguchi et al. (2011), Önöz and Bayazit (2012), Yue et al. (2002), and many others. Even the Spearman's rho test have various variants including Hamed (2016), von Storch (1999), etc.The important point here is that you need to realize that there are uncertainties in trend analysis and therefore the need to take into account the possible influence of the choice of a particular trend test on the results of analysis (see Onyutha 2016 for details).
Please, you will find that the references below are in different styles but I am hopeful that it is not the correctness or order of the references which will matter.
Check some important links at the bottom of my answer to your question.
REFERENCES
H. von Storch (1999) Misuses of statistical analysis in climate research,” in Analysis of Climate Variability: Applications of Statistical Techniques, H. von Storch and A. Navarra, Eds., chapter 2, pp. 11–26, Springer, Berlin, Germany.
Onyutha C (2016) Statistical Uncertainty in Hydrometeorological Trend Analyses. Adv. Meteorol. 2016, 1-26. https://doi.org/10.1155/2016/8701617
Mann HB (1945) Nonparametric tests against trend. Econometrica 13(3):245–259.
Onyutha, C. (2021) Graphical-statistical method to explore variability of hydrological time series. Hydrology Research (2021) 52 (1): 266–283.
Sen, Z. Innovative trend analysis methodology. J. Hydrol. Eng. 2012, 17, 1042–1046.
Spearman, C. (1904) The proof and measurement of association between two things. Am. J. Psychol., 15, 72–101.
Lehmann, E.L. (1975) Nonparametrics, Statistical Methods Based on Ranks; Holden-Day Inc.: San Francisco, CA, USA.
Sneyers, R. (1990) On the Statistical Analysis of Series of Observations; Technical Note no. 143, WMO no. 415; Secretariat of the World Meteorological Organization: Geneva, Switzerland, 1990.
S. Yue, P. Pilon, B. Phinney, and G. Cavadias (2002) The influence of autocorrelation on the ability to detect trend in hydrological series. Hydrological Processes, vol. 16, no. 9, pp. 1807–1829.
B. Önöz and M. Bayazit (2012) Block bootstrap for Mann-Kendall trend test of serially dependent data. Hydrological Processes, vol. 26, no. 23, pp. 3552–3560.
K. Noguchi, Y. R. Gel, and C. R. Duguay (2011) Bootstrap-based tests for trends in hydrological time series, with application to ice phenology data. Journal of Hydrology, vol. 410, no. 3-4, pp. 150–161.
S. Yue and C. Wang (2004) The Mann-Kendall test modified by effective sample size to detect trend in serially correlated hydrological series. Water Resources Management, vol. 18, no. 3, pp. 201–218.
K. H. Hamed (2009) Enhancing the effectiveness of prewhitening in trend analysis of hydrologic data. Journal of Hydrology, vol. 368, no. 1–4, pp. 143–155.
K. H. Hamed (2008) Trend detection in hydrologic data: the Mann-Kendall trend test under the scaling hypothesis. Journal of Hydrology, vol. 349, no. 3-4, pp. 350–363.
K. H. Hamed (2016) The distribution of Spearman’s rho trend statistic for persistent hydrologic data. Hydrological Sciences Journal, vol. 61, no. 1, pp. 214–223.
For uncertainty in trend analysis
Article Statistical Uncertainty in Hydrometeorological Trend Analyses
The new trend test
Article Graphical-statistical method to explore variability of hydro...
MATLAB-based tool (CSD-VAT) for applying the new method
Code CSD-VAT: CSD-based Sub (Trend) and Variability Analysis Tool
Other relevant papers
Article Statistical analyses of potential evapotranspiration changes...
Article Identification of sub-trends from hydro-meteorological series
Article Trends and variability in African long-term precipitation
Article Trends and Variability of Temperature and Evaporation Over t...
This paper takes a different approach by simply counting the number of exceedances of particular depth-duration-frequency (DDF) combinations for the Semi-Arid Southwest region, and for the the Ohio River Basin and some adjacent states, in the U.S. Those numbers are then plotted and analyzed with interesting results. The DDF threshold values are taken from NOAA Atlas 14, Precipitation-Frequency Atlas of the United States, Volumes 1 and 2.
The paper is aimed at hydrologic analysis which is quite different from the more common climatological approaches and definitions. However Appendix A.3 in those NOAA Atlas 14 volumes assesses stationarity by examining trends in the annual maximum series, and shifts in the mean of the annual maximum series, which were extracted, quality controlled and used in the production of those NOAA Atlas 14 volumes.
Bonnin, Geoffrey M., Kazungu Maitaria, and Michael Yekta, 2011. Trends in Rainfall Exceedances in the Observed Record in Selected Areas of the United States. Journal of the American Water Resources Association