It sounds like there's an ungrounded (hot) conductor shorted to ground somewhere. Finding the short is likely going to be a tedious task.
Check your work
Did you install grommets or clamps, where the wiring passed through knockout holes in boxes? If not, start by inspecting the wiring where it passes through the knockouts. The edges of these holes (especially on thin recessed lighting boxes) can be quite sharp, and can easily damage cables and wires as they pass through the hole.
Check all your connections, making sure nothing came into contact with ground or non-current carrying metal. This can sometimes happen when wires are pushed into boxes, so check for loose or pulled out wires.
Inspect cables/wires where they are supported, especially where supported by metal staples or clips.
Check existing work
Start by turning off each breaker in the panel one at a time, while monitoring the voltage on the drop ceiling rails with your non-contact voltage tester. When the voltage goes away, you've found the offending circuit.
Once you know which circuit is causing the problem, it's time to start the tedious task of locating the fault. Do your best to trace out the circuit, and determine all the equipment on it. Open up all outlets and equipment, and inspect the wiring inside.
It may help to move along the circuit, and disconnect the load conductors (the conductors feeding the next junction) at each junction. Then energize the circuit, and check for the voltage again. This can help you isolate the problem, and will help you focus on the problem wiring. Once you find the offending part of the circuit, methodically inspect the wiring. Checking for damaged wiring, loose connections, improper wiring, etc.
Unless you get really lucky, you're likely in for a long tedious search. Good luck.
As-wired, the three wires are Hot, Hot and Neutral. You do not have a ground wire, (unless this run is in metal conduit, in which case the metal conduit is the ground path.)
A 120V receptacle has 3 wires: Hot, Neutral and Ground. Your conceptual problem is you are trying to match 3 wires to 3 wires. And that will never work because those 3 wires have different functions.
First, sort out the Ground question.
Look closely at the conduit carrying the wires. I assume it is conduit because I see 3 loose wires same color. If indeed the conduit is all-metal, and continuous, and joints in good condition, then you have a ground *in the conduit" which should be accessible at the metal box attached to the conduit. It should have a hole tapped 10-32 that you can put a grounding screw into.
You might have an electrician test this believed-to-be-ground path to see if it can really carry a strong current flow. To be actually useful, it must be able to flow enough current to assure the circuit breaker trips.
If you do not have a viable ground path, or if you want "belt and suspenders", you can retrofit a ground: route a copper ground wire from the shed back to the house, via any legal method. If it is indeed in conduit, you might be able to just fish (or even push!) the ground wire through.
The last option is to simply leave it ungrounded. Toss out those NEMA 5 (3-prong) receptacles and buy NEMA 1 (2-prong) receptacles. This will be allowed if the wiring is old enough to be grandfathered (i.e. was installed before grounds became law). If you do this, I strongly recommend you replace the circuit breaker with a GFCI type.
Here's what you can't do: Grab some green tape and mark one of the hot wires as a ground. It is now illegal to remark a black wire (of this size) as a ground. A ground must be green, green/yellow stripe, or bare.
A local grounding rod is no substitute
If this were any more sophisticated, you would need a local grounding system (rods) at the shed. You don't because this is a single (multi-wire) circuit into a single breaker. Here's the point: Do not use a grounding rod as a substitute for a ground wire. You need a ground wire back to the panel so that ground faults will trip the breaker. A ground rod won't do that because dirt is just dirt, it is not a magical superconductor. Dirt conducts electricity about as well as you'd expect, and it won't flow enough current (if it flows any at all) to give you a reliable breaker trip.
So don't even think about using a local grounding rod at the shed as a substitute for running a proper ground. It is a nice add-on, though: what it does is give a very direct path for Mother Earth's electricity - lightning and static electricity - to get back to its source.
Then sort out the Neutral
I see in the service panel where one of the black wires has white tape on it. Similar to green tape, it is now illegal to re-mark a black wire (of this size) to be a neutral. If the installation is old enough it may be grandfathered, and I would take the assumption that it is. In that case, it's OK to use new tape to freshen the mark, as that one is falling off.
It's important the mark be intact on the other end of the wire too. Freshen it there too. It's vital not to get the wires mixed up.
You have Hot-Hot-Neutral. Use it that way.
What you have is called an Edison circuit or a multi-wire branch circuit. This can give you 240V (hot-hot) or 120V (hot-neutral) or 120+240 (hot-neutral-hot) for dryers or subpanels which use both.
If you want a 240V circuit, use the two hot wires. This could go into a NEMA 2 receptacle (or if you have a good ground you can wire, use NEMA 6).
If you want a 120V circuit, use one of the hot wires and the neutral. Do not use the other hot wire. Without a ground use a NEMA 1 receptacle, with ground use a NEMA 5 as in your picture.
If you want a 120+240V circuit, use both hots, and neutral, and ground. If you have ground, use a NEMA 14 receptacle. If you do not have ground, do not install it. 120+240 without ground kills people.
Best Answer
You are being written up for a missing Equipment Grounding Conductor (EGC) which is the ground wire that goes with hots and neutrals in every circuit or feeder.
The thing between this meter-main and your house is feeder not service wire. Above the meter is service wire. Feeder requires EGC, service does not.
What I see
is 200A service coming from the top.
I see a bare wire that is part of the Grounding Electrode System which bonds the meter-main chassis itself to local grounding rods. However this is not an EGC ("ground wire") and does not go anywhere an EGC needs to go.
I see the main feeder going into the lower pipe in 3 wires. I see no ground wire (EGC), so yeah, the inspector got it right.
Fire your electrician. What were they thinking?
If the power company installed this feeder, that would be why. They follow a different codebook (NESC) and they're not used to working in NEC rules. They make this mistake all the time.
Some people think that if the pole has a ground rod (GES) and the house has a ground rod (GES), they don't need a ground wire (EGC) also, because there's dirt between the two. Um... dirt doesn't conduct electricity that well :)
What to do
You will need to add an appropriate sized ground wire to the pipe. If the hot wires are exactly #2/0 Copper or #4/0 Aluminum, you can use a #6 Cu or #4 Al ground wire. (However, if the hot wires are larger than that, the ground must also be larger, forcing you to #4 Cu or #2 Al).
The added wire can't exceed conduit fill limits. The ground wire can be bare if it's copper, which helps with conduit fill. It's OK to use aluminum hots + copper ground.
You can fish just the wire if you can successfully get a fishing tape down that conduit with the wires in there. If not, you'll have to pull the 3 wires and then pull them back in with the 4th wire.