I put excess reserves in quotes because I know that in many countries there are no reserve requirements, so what I really mean by "excess reserves" is reserves in excess of what banks like to maintain for the purposes of making settlements between each other (is there a special name for this kind of excess?). My hunch is that they have used them to buy bonds or shares from non banks, creating new demand deposits in the process. Or have they mostly left those reserves in their accounts at the central bank?
EDIT: I know that reserves can not be used directly by a bank to purchase something from a non-bank. But reserves enable these purchases because they may be needed in settlement of the purchase.
EDIT: Given the nature of the answers so far, perhaps it may clarify this question if I give some hypothetical answers...
- The banks could have said to themselves "Perhaps, before 2008, we had underestimated how much reserves were really needed, the new reserves from QE were actually required after all and not 'excess' in any way. Let's just leave them in place and collect the interest on them that the fed pays"
- Having these new excess reserves allow us to purchase more bonds and/or shares from non banks. Let's buy some.
- Having these new excess reserves allow us to make more loans than before, so let's increase our lending.
Given that there was so much uncertainty about what the effects of QE would be when it started I have no doubt whatsoever that research will have been done into exactly how banks employed the extra reserves.