The description of the Melpomene hat reads: "participate in successfully closing or deleting a question".
I've found that it is awarded generously: you don't have to have be able to cast close votes or delete votes, it is enough if you flagged a question that is later put On Hold.
However, it is not awarded for flagging spam.
At first I thought that maybe spam was removed so fast and permanently from the system that it is gone before the Winterbash hat-awarding script runs.
But there was a spam post on WebApps.SE that lasted for 30 minutes before it was gone, and I didn't get the Melpomene hat for that.
So the question is - is this by design? Does flagging spam not count for this hat?
For the record, I don't mind not getting it on WebApps.SE. I've got it on enough other sites. I'm just curious.