It's not clear what exactly are you doing and trying to accomplish.
You really need to provide more details, and definitely a schematic, maybe a photo, screenshot of the scope signal and the purpose of your circuit if it's not clear from the schematic.
My guess is that you are driving a 110V AC load a square wave inverter, but I could be wrong because you have left plenty for us to imagine.
It seem that what you have is a "leading edge spike".
Such spike is usually easily solved with an RC integrator.
simulate this circuit – Schematic created using CircuitLab
You can experiment with the capacitor values from 1nF to 100nF, start from the lowest value and change up until the spike is gone.
EDIT: Now that you have provided images and details, it is more clear what is happening, and it is due to square wave supply as I suspected, or modified square wave, to be more precise.
The spikes are happening on, and are caused by, the rising and falling edges of the voltage supply. It is really not a mystery why this is happening, as those edges represent extremely fast transitions and are bound to cause spikes even at very low inductances such as that of a straight piece of wire. Another source of the spike could be due to the electro-chemical contact reaction between the probes and the soil, where a sudden change in voltage or current can't be conducted fast enough through such "galvanic cell" (but I could be wrong).
The spikes happening when the load is disconnected (no current) are most likely due to the wires acting as antennas for those very fast transients. Disconnecting the top lead next to the generator and then scoping across the shunt should verify this.
The wire inductance problem can be reduced by using twisted wire pair instead of just regular wires. Basically, treat the problem as if you are working with radio frequencies running through the wires.
The spikes across the shunt and the probe may be solved or reduced by placing 1nF to 100nF capacitors across them (one across the probe, another one across the shunt); start with 1nF and gradually move up until the problem is eliminated or significantly reduced.
Another solution would be to change your source voltage into a sine wave by using an LC low-pass filter (set around 60Hz) or an LC resonant circuit across the supply (set to 60Hz). This would get rid of your sharp transitions which cause the spikes.