Have you tried plugging the equipment into a different wall socket ? Maybe the "monitoring" power line (or that specific socket) can't handle the temporary load increase you have when plugging the equipment, so the voltage momentarily drops, making the usb charger drop the power to the arduino long enough to have it reset.just my 2 cents.
Quick possibility is add large external capacitor on the 5V from the adapter. Say 4700uf at 10V or more...
So the Arduino + wifi are working when nothing else is plugged into the AC outlet, then something else gets plugged in and wifi stops functioning, correct? Can you post a parts list? Which Wifi-shield? (datasheets are helpful too!) and a description of how you have connected everything (schematic).
So you have multiple Arduinos setup with essentially the same code and the act of plugging in the monitoring equipment puts the arduino on the fritz, correct? Or multiple devices connected to the Arduino? If it's the former, we can probably rule your code out as the direct source of the problem.