Correlation between two ordinal categorical variables What is the best statistical test for investigating if there is any correlation between 2 categorical variables?
Both are satisfaction scores:
1st variable is:

Overall satisfaction with the service.
1: Not at all satisfied; 10: Completely satisfied

2nd variable is:

Satisfaction with the availability of information for the service"
1: Not at all satisfied; 10: Completely satisfied.

 A: I would go with Spearman rho and/or Kendall Tau for categorical (ordinal) variables. 
Related to the Pearson correlation coefficient, the Spearman correlation coefficient (rho) measures the relationship between two variables. Spearman's rho can be understood as a rank-based version of Pearson's correlation coefficient.
Like Spearman's rho, Kendall's tau measures the degree of a monotone relationship between variables. Roughly speaking, Kendall's tau distinguishes itself from Spearman's rho by stronger penalization of non-sequential (in context of the ranked variables) dislocations.
A: Both of these have enough levels that you could just treat them as continuous variables, and use Pearson or Spearman correlation. You can then calculate a significance (p) value based on your correlation and sample size.
If you really want to treat the data as categorical, you want to run a chi-squared test on the 10x10 matrix of overall satisfaction vs. availability satisfaction. You will need a decent amount of data for this (~thousands), since the majority of the cells should contain at least 5 observations for the test to be valid. This would allow for more general types of dependence between the two measures, in which even nearby levels show different relationships (e.g. rating1=9 tends to predict rating2=4, rating1=8 tends to predict rating2=10) which are probably not likely in your data.
A: I went and searched for it, found this from John Ubersax:   http://www.john-uebersax.com/stat/tetra.htm
and some papers
https://link.springer.com/article/10.1007/s11135-008-9190-y
https://escholarship.org/content/qt583610fv/qt583610fv.pdf
