Excelent question with a difficult answer. In fact the concentrations of miRNAs in serum or plasma range from nanomolar to femtomolar. Depending on the miRNA in which you are interested, the limit of detection will change.
You have also to take into account the volume of sample that you must use. In humans the classical detection method by qPCR will need around 200/300 ul of serum to have enough RNA.
Thanks for the insights Paco, very useful on several fronts. The idea of PoC testing for miRNAs is relatively new so it will be interesting to see if any of the emerging approaches eventually match or surpass qPCR.