After our last discussion I though I would collect some data on the % of items actually upgrading. Since then I've done 250 runs and looted 500 items from Rezan and here are the results:
Total Single Upgrades: 90 items have upgraded with only Socket or a Tertiary stat, or 18.00%.
Total Double Upgrades: 7 items have upgraded with both Socket and a Tertiary stat, or 1.40%.
Now we can go a bit more specific and loot at the individual type of upgrades:
41 items upgraded with only Socket, or 8.20%.
22 items upgraded with only Avoidance, or 4.40%.
13 items upgraded with only Speed, or 2.60%.
8 items upgraded with only Leech, or 1.60%.
6 items upgraded with only Indestructible, or 1.20%.
3 items upgraded with both Socket and Leech, or 0.60%.
2 items upgraded with both Socket and Indestructible, or 0.40%.
1 item upgraded with both Socket and Avoidance, or 0.20%.
1 item upgraded with both Socket and Speed, or 0.20%.
What we can see from this is that Socket is by far the most common upgrade and that getting a socket is almost as common as getting any of the Tertiary stat upgrades.
This data is by no means perfect since the sample size is very small, but I though it was better than nothing, I will update it once I get my Shield. I should also note that only 6 our of 8 of Rezan's drops can have Indestructible, which is why it's the rarest Tertiary stat.
PS: This test might only be true for BFA Dungeons, I have no idea if the upgrade chance is the same for other expansions or other drops.