Dave Jones's uCurrent - oscillations
I tried many things, changing ICs, changing caps, etc.
Finally I found this thread:
https://www.eevblog.com/forum/blog/eevblog-1057-current-murphy/

Dave found the culprit. It is not the output stage opamp. Problem is in the opamp which makes virtual ground. I have used LMV321 from onsemi and Dave had the same issues with that.
https://youtu.be/1VlKoR0ldIE

There are 2 solutions for that:  


change onsemi's LMV321 to TI's one  
add 100nF capacitors from V+ to virtual ground and from V- to virtual ground (in my schematic from VDD to VGND, and from VSS to VGND)