Here's an article that seems to indicate yes:
https://www.quantamagazine.org/the-unpredictable-abilities-e...
I don’t know but I’m in disguise as a paperclip just in case.
Has there been any indication that it's better at designing models than its designers?
No
What's the singularity?
Technological singularity is very precisely located at the point in time when technology can / could continue to exist and develop without humans. This is clearly not the case right now.
The invention of LLMs is surely closer to that hypothetical point in time than the invention of steam machines or iron swords, but even nowadays technology is still very dependent on a working human society to continue functioning.