Depends on your budget, but the JouleScope [1] can measure down to half a nanoamp. Absolutely amazing piece of kit for the price ($1,000).

They also have a python library [2] so you can do lots of stuff inside Jupyter.

[1] - https://www.joulescope.com/

[2] - https://joulescope.readthedocs.io/en/latest/

I’m guessing professional EEs have better tools but in my hobbyist experience the power profiling kit II works well for analyzing low power (0-5v) circuits. The software tool provided by Nordic is fairly capable and easy to use. At $89 from DigiKey it’s sort of a no brainer to pick one up to play with.

I would recommend the UM34C USB tester, cheap, good, has BLE connectivity and a 3rd party open-source python app[1] to log and monitor the data.

[1] https://github.com/kolinger/rd-usb