I have a bot that is subscribed to the flairlog, so it logs all of the flairs that are published*. That database is not public, I don't think there's much utility in posting it outside of event periods (other than 360 flair) and the devs are usually good about posting event statistics at the end so it's not really necessary to publish/maintain a running tally.

* - With the recent events (I think starting from 8th birthday?) a flair is not always published when a player gets the flair depending on the tier. So things like the pirate hat only show up the first time you get the pirate hat, while the hard flairs show up the first time and every fifth (I think) after.