There is no best method in LST retrieval from Satellite.Think about the availability of the type of Thematic Mapper in your such as TM+, split window or mono window algorithms. It is first important to know whether the intended data are available (do data search first). Divide your year into different months of their seasons such as Summer, Winter, Fall, Spring etc. Take at least one month in each season across the year and the years of the study.
I am agree with Dr. Nwaerema. Furthermore, to select a specific month the faster way to do that is to analyse the cloud cover of two or three years and select the month with less cloud cover. The best but tough way is to extract the clouds cover mask of every month and select the month based on this data. I think most of the satellites has this information included in the images, try to extract them using a loop. (e.g in Landsat MTL.txt file has CLOUD_COVER = 0.00 and CLOUD_COVER_LAND = 0.00
I agree with @Peace. Data search is an important part of the work. Proper data search will help you assess the images that you can use. Earth explorer gives you different views of the image (thermal and visible), it also gives you details about the data quality and cloud cover. With regards to the best month, it depends on the focus of your work; do you intend to do a seasonal analysis or you just want to assess the land surface temperature over the area within the past 30 years? It is important to take into consideration the climate of the region of interest.
Many researchers have compared the results of different LST retrieval methods and the accuracy results of these studies showed that the best method changed case by case. For 30 years evaluation, it depends on you to choose the month regarding the context of your study. For example, if you want to evaluate seasonal changes you should choose at least one image for each season in a year as stated by Peace Nwaerema and it would be better if the acquisition month of an image is same for different years.
In another condition, you can analyze images acquired on the same month of different years. In other words, you can download images that are acquired in July for last 30 years and analyze them statistically. But, please pay attention to apply radiometric and atmospheric corrections for temporal analysis.
Peace and Daramola provided detailed answer to your question. However on the issue of the best month, you need to consider the local climate of your research area: as what is best in a certain month over a certain study area may not be the best over another study area in the particular month.
.....use the same season throughout possibly same month if quality images are available....after you have identified your local climate patterns as Mr Emmanuel earlier suggested.
Thank you for your good suggestions Peace Nwaerema Israel Ropo Orimoloye Emmanuel OlaOluwa Eresanya Aliihsan Sekertekin Mojolaoluwa Daramola Salomon César Nguemhe Fils Ivan Lizaga