Max current and wire gauge
Attempt to measure mV-range signals under the 24-V bias with a 8-bit scope and individual (and therefore different) probes is a failing proposition. There will be too much noise. There are basically three options:


Use a dedicated active differential probe, but they are pretty expensive. To accommodate 24V offset you would need a TDP1500 probe (+-25V, $4K price), or high-voltage probes like P5200A (eBay ~$700);
Built your own sensor using something like INA199 or ZXCT1107/09/10 from Zetex'



The above circuit is literally all you need, an IC, and a scaling resistor. You might need to calibrate this circuit with known stable loads before taking actual measurements.


put the shunt resistor into ground wire as suggested by "winny", if the load allows this small 20-30mV bias. Subtracting readings from two different probes still be subject to general uncertainties of low-volt measurements.


The answer to sustainable or impulse capability of a copper wire is given here, on stackexchange.