First those readings can be quite normal and very common, we must remember that the earth is in parallel with the neutral of this service and current will take all paths back to the source (transformer), the neighbors service the its grounding as well, even a ground rod of 50 ohms will allow 2 amps of current to flow, but if you turned off all A leg breakers as well as any two pole breakers then measured the voltage on the B leg with some heavy loads turned on if it should drop below 120 volts on one leg and goes over 120 volts on the other leg then you could have a secondary neutral problem, a Small voltage rise and fall can be normal if there is some voltage drop on the service conductors, also if there is a water pipe ground connection and the street side of this connection is copper or iron pipe then it can be also masking a bad neutral, remember to shut off the main before turning off all the A leg breakers as if you do have a bad neutral you could possibly over voltage equipment in the house if you start turning off breakers to force an unbalance condition with the main still on, also keep in mind that disconnecting any grounding can increase the voltage on the services grounding to earth so you want to make sure that no one at the house is in a position to touch anything grounded to the service even the water pipes or valves, also if you do this with all the breakers on both A and B legs turned on you could risk damaging the customers equipment from over voltage, always turn off the main and turn off all the breakers on one leg of the panel before disconnecting the any grounding to unmask a parallel neutral path that could be masking the neutral problem.
Also keep in mind that if you force an unbalanced condition on a transformer that feeds other houses also, if the bad neutral connection is at the transformers X0 bushing (neutral) you could also be causing over voltage in these other houses because they will still have loads on the leg that will go higher, I would not fully load one leg at first, just load it a little at a time then check each leg to neutral voltage before going any more to see if you start dropping voltage on the one your loading and rising the voltage on the other unloaded leg, if the unloaded leg reaches 125-130 volts while the loaded one goes down to 115-110 volts then stop there and check to see if this voltage imbalance is also at the meter on the line side of it, if it is then call the utility, these types of problems must be done in steps to make sure you don't cause other damage from over voltageing other equipment in this house or any neighbors that might share the same transformer.
Another test I do when I find a little current on the service grounding is to test for voltage from the service grounding or even the meter can to remote earth by sticking a screw driver or short rod into earth about 20' from the service and run a wire to the meter so I can use a DVM or other high impedance voltage meter (not a wiggy) to see if I have any voltage over 10 volts between this earth test rod which is the voltage that tells me that there maybe a primary neutral to the transformer connection problem, although not a very common problem it does happen, generally the current will be low but the voltage can be high, a secondary neutral problem is a higher current but the voltage is normally low depending upon how well the service is balanced, but when you start seeing over a 10 volt reading but a low current then a primary neutral to the transformer flag rises because this could be very dangerous, also never disconnect any grounding if this is ever discovered because you could end up having the primary voltage on the grounding of this house as the transformer tries to return its current through this path.
If you ever suspect a bad primary neutral to a transformer connection from measuring the voltage to a remote earth point then ask the utility to test the primary neutral at the transformer to make sure it is returning its current to the MGN, they do this by using a HV clamp on amp meter to make sure the transformer is returning its current through the correct wire that connects to the MGN, some utility's don't use a WYE local distribution system so in these cases this would not happen, but many do like ours that uses a 12400/7200 volt WYE system for local distribution.
another good sign of a bad primary neutral to transformer connection is if the customer has experienced and underground water pipe leak problems and they had a plumber make repairs with PVC to stop the problem or had a dielectric union installed after the leaks were repaired, also after this is done you start seeing higher voltages to earth readings many times in the 20 volts or above,
