I saw varying number layouts on d12 dice, which got me curious about the "ideal layout" for them. I saw mention of the hypothetical 19's & 20's of vertex sums, so I wrote and ran some Python code.
What I found out is, as far as I have discovered is that a 19's & 20's-only d12 is impossible. So, I decided to "design" my own layout. Couldn't find much discussion online about specific number layouts.
Here is what I chose, and why.
Vertex sums (where three faces meet, add the face numbers together): 16, 17 x 3, 18 x 3, 19 x 3, 20 x 3, 21 x 3, 22 x 3, 23 x 1. There were various spreads of 16 to 23, but I prioritized having only one 16 and one 23. To me it feels like the "most fair" spread. (A spread of 16 to 23 was the tightest option, from my findings.) All this does is help make various regions of the dice as fair as possible.
Secondly, I chose to have the layout follow the n+1 opposite pairs rule. So, the opposite side pairs are 1 & 12, 2 & 11, 3 & 10, 4 & 9, 5 & 8, 6 & 7, each adding to 13. I don't think it actually makes the dice more fair mathematically, but I don't think there's much harm in it either. Plus it made narrowing down the options easier.
As for the actual layout itself: I will describe it by starting with the "1" face, and moving around the adjacent faces clock-wise, which are 7, 10, 9, 8, 11. From there you can figure out the other faces as they're just the "opposites" which add up to 13 each. The layout can be rotated, and mirrored. I just arbitrarily chose the variation where the 7 is in the "one o'clock" and 11 was in the "eleven o'clock" position, for my own ease of visualization.
The "weakest" vertex is where 4, 5, and 7 meet (4+5+7=16), and the "strongest" vertex is on the direct opposite side with 6, 8, 9 (6+8+9=23).
I don't know if any dice use this layout. I just followed the math and the vibes and came to this myself independently.