Why is mRNA used as a biomarker for cancer over tRNA or rRNA?
mRNAs encode specific genes and perform specific functions whereas the other two perform housekeeping functions in all cells. Changes in their expression are related to global changes in the cell.

The expression of rRNA and tRNA is affected in certain conditions such as stress and also possibly cancer (may be due to high replication rates). However, even if they are affected in cancer you cannot use them as markers for specific cancers as they are present and highly expressed in all cells. 

Therefore specific tumour suppressor genes and oncogenes are studied.