Why is ground voltage non-zero, and why does it increase when I plug in an LED?
In Breadboards with lengthy cables a 50 mV offset is acceptable and is not an issue. If it is affecting measurements, then you should consider shortening the cables and possibly use thicker cables. a 0.5 ohm resistance cable with 5mA current will give 2.5 mV drop. If there are other currents through the ground cable, assume Entire board current, then the drop will go up. For ex. 100 mA current now will cause a drop of 50 mV.