Which correlation method to compute the correlation score between different clusters of Sc-RNAseq data?
Disclaimer: This is not a complete answer but more a comment, posting as an answer because of character restrictions for comments. Also, I am not a statistican but a molecular biologist who did quite some reading/courses on stats.

Two scRNA-seq tools, namely SingleR and RCA (Reference Component Analysis, Li et. al, 2017) compute correlations of bulk transcriptomic data to single cells; for this they use Spearman correlation and Pearson correlation respectively (The two tools use the same bulk transcriptome reference). I am using these two tools for different purposes so I did not really compare their results, however, single cell clusters that I compute using the reference components (calcuated with RCA) is in good agreement with single cells labels computed with RCA (adjusted rand index of 0.81).

In addition to that, please take a look at scMatch: a single-cell gene expression profile annotation tool using reference datasets, they claim (scMatch computes bulk - single cell correlations like SingleR and RCA):


  From this comparison we found Spearman’s correlation coefficient
  outperformed Pearson’s correlation for all six libraries.