Uh. Considering the aarne-thompson classification system is classifying memes...
... as a more general thing, not that I've heard of. I wouldn't be surprised if someone in academia has tried, though, probably in some kind of meta-literature/communication/etc. program (study of the study itself and how it's organized, rather than whatever the field is focused on), at least assuming I'm not misremembering the correct prefix for that sort of thing (which is entirely possible, mind you). The more public/casual attempt would probably be projects like TVTropes and the few other sites of similar nature.
The problem with making one that covers memetics as a whole is that at the core of them memes are incredibly broad -- they're basically just concepts related to communication and media that have been identified as sufficiently discrete to stand on their own in some way or another and to varying degrees. Modern discussion, particularly casual discussion, is largely focused on internet centered stuff, but memes are a lot older than the internet, we've just started using the terminology relatively recently (inspired by attempts to apply concepts related to genetics to human communication, near as I can recall). Generally you have attempts to classify a subset (such as folklore, yes) instead of all of it in one go, because all of it in one go would be trying to make a classification system for pretty much the entire spectrum of human communication. Which, uh. Is a little daunting, ehehe.
People have been working at it but they tend to break things up, and by the time someone's done(ish) with one thing they're usually too old or invested to go back to trying to integrate what they've done with everything else, and probably either them or their colleagues are highly resistant towards letting the field of study they've developed get folded back into some other one. All of which makes integrating everything... slow. Given the time frames involved (we've only really been doing this sort of thing for a century or three), it's almost certainly going to be a while before you see a full framework of some sort starting to be pieced together. In the mean time, you'll just want to identify specific field equivalents and peruse as appropriate, if you're looking for that sort of classification system.