This is the TO-92 cased LM317L version that only can handle 100mA max.

It still requires a worst-case minimum of 2.5mA load for proper operation.
Thus, as you noted, operating at 50µA doesn't work properly.

 

Sounds interesting and hadn't heard of them before. Looking at the PDFs for the 1N52xx and 1N53xx they run from nominal 0.22mA to 4.7mA if used simply as a current limiting device. Unless there is some other adjustable current ciruit method? No typicals seen in the PDF or found googling.

Checkout Central Semiconductor
https://my.centralsemi.com/paraSear....php?task=bldPartTable2&PGroup=10&subGroup=50

 

I changed the load on the LM317L to 200kΩ and increased the Vs to 11.57V to trim the current output. Over 6.5 hours it has very slowly drifted from 50.001uA down to 49.976uA. So, increasing the load and drive added a great deal of stability! It is flitting between 49.978 - 49.980uA as I write this so appears to be going back up.

 

Yeah I saw that on the pdf and wasn't even sure it would actually get down in the uA range but it does. This is the TO-92 cased LM317L version that only can handle 100mA max. Just not very stable and with only 1.5V driving it. But I tried and did learn something from it... I did put a 47k resistor on the output as a load and the Vs had to be 3.8V to drive 50uA but still unstable.

Which says it is not regulating at all. All you have done is adjust the voltage until you get 50uA. You could do that without the LM317.

 

I changed the load on the LM317L to 200kΩ and increased the Vs to 11.57V to trim the current output. Over 6.5 hours it has very slowly drifted from 50.001uA down to 49.976uA. So, increasing the load and drive added a great deal of stability! It is flitting between 49.978 - 49.980uA as I write this so appears to be going back up.

As noted, that is not working as a current regulator, since a current regulator gives a constant-current independent of load or supply voltage, and that circuit is doing neither.

To repeat, it is not possible to use the LM317L as a 50µA current regulator.
So why are you still trying?

 

Curiosity...

 

Curiosity...

Okay, I understand that.
But a big part of doing something out of curiosity is understanding the results.