I can't see what's going on there. Which one of the circuit is tripping, outlets or lighting? Go as far back as you can and disconnect that section at the J box and see if that trips the breaker. If it does not, then you know its downstream. If it still does, then its up stream. It can be several things like a nail through the Romex, roden, or a cut on the wire at the metal connectors. Did you use metal boxes?

There are mostly outlets on the circuit. There is one switched overhead lighting fixture. The "gray stuff" I was referring to earlier is the direct burial style of Romex. It's in PVC conduit. I don't think it could be nail damage, since all exposed Romex is in EMT conduit, and there haven't been any construction projects near the circuitry. There is one branch of the circuit that was added later that services a couple of outlets, but that circuit has worked fine for a couple of years. I think I'll disconnect that one first and see what happens. A project for daylight tomorrow...

Oh, and metal boxes were used indoors; weatherproof PVC outdoors.

It would appear this installation is below code requirements - beginning with the lack of GFI equipment. The wire nutting of a bunch of neutrals together should also likely result in red tagging. However the correct installation and isolation of the neutral and grounding bus bars for subpanels is often missed by professional electricians. See: http://www.creia.org/images/stories/sub-panel-grounding.pdf

Sounds like it's time for a rework by someone who is up to speed on the code.

Check those Romex connectors in the first picture on the second page in Jim's link above. Sometimes they are cut there because of over tightening. I think something might have come part in one of the J boxes.

Shut the light off and turn on the breaker see if it still trips.if it doesn't its the light circuit.If it does leave someone in the room in the dark turn on the breaker and see if there is an arc when it trips they will see where it's general location is ,maybe hear it too. Start pulling out the outlets see if you see any burn marks ( power off ) generally hot wire wire hitting metal box some where.

Thanks for the suggestions. I'll check the j boxes when I get home tonight. I can hear the arcing in my barn, and I have an idea about where it is. I'll disconnect that branch line tonight and see if it solves the problem.

You could have a bad circuit breaker. Borrow a clamp-on ammeter and measure the current.

I don't believe the burned common is the problem, that should not cause your circuit breaker to trip unless it is causing a heavy starting load to cycle repeatedly (like an air compressor or a fridge).

If there is no visual damage then you need a megger to do an insulation test.

A bad circuit breaker was my first thought, since the breaker was tripping with zero or minimal load on the circuit. I've already replaced the breaker, with no effect on the problem. The entire circuit is generally subjected to minimal loads - a couple of fluorescent lights (corded, that are plugged into outlets), and battery chargers. There is an air compressor that is used occasionally, but it's never tripped the breaker. Last night, after I unplugged all loads from the circuit, and reset the breaker, it tripped after 60-90 seconds. Very strange...

There are three other breakers in the sub panel. None of those has ever tripped.

You need to start disconnecting loads/lines from the circuit until it doesn't trip. Start at the circuit breaker itself, and then work towards the end.

Are you sure this is just a circuit breaker, not an arc fault or ground fault breaker?

You must have a cut wire somewhere that grounding itself. Now its just a process of elimination.

Pretty sure it's just a circuit breaker, as only one is affected in the sub panel. I agree that the process is to start disconnecting the branches on the circuit one by one until it stops misbehaving. That process will begin tonight... ;)

If you can "hear" arcing noise from somewhere in your shop I doubt it is a bad breaker, Substitute a known good breaker (of same amp rating) from another circuit and see if the problem moves with the breaker.
I had a similar situation develop with a circ running my compressor It would spark once in a while and pop a breaker. Turned out to be a piched wire behind it which only shorted to the compressor when running (and vibrating)

I wish it was that easy, I really don't think its a bad breaker. LIke John said, swap another one in there and see. Simple and easy if you have no testing tools.

Turn it on and listen for the hissing,general area and start there.Look in the dark for arching,Like looking for bad spark plug wires at night. Start there. Sounds like a wire is grounding but not a dead ground.look for burnt wires. Look ,listen,smell. LOL!

Sorry. Last night I meant to say that I was pretty sure it was NOT a bad circuit breaker, as I've already replaced the offending one with a new one. I disconnected the "prime suspect" branch last night, but that wasn't the problem. I'll start on the others tonight. If that doesn't cure the problem, then I'm guessing that the issue would have to be in the 6-gauge wires that bring the power from the sub panel to the various buildings. We do have gophers, but have they been known to chew through underground PVC conduit?

Problem cant be in your 6 g supply wires. The circuits should be set up with a breaker protecting the 6 g wires at the supply panel (60A). You should have 20 or 15 A breakers protecting your branch circuits (depending on wire size either 12g or 14g. Breakers protect the wire downstream not upstream at the supply.
When you say you disconnected the branch but it wasn,t the problem .How did you determine that? With no load connected the breaker wont trip.
I would look for the source of your crack /snap sound in the other room and start there.

I disconnected the branch circuit at its junction box, then reset the breaker to "on." It tripped within about 60 seconds. I could hear the noise in the barn (where a couple of the junction boxes are), but didn't see any visible arcing (it was pitch black in there). When I was speculating about the supply lines, maybe I should have said that the 6 (or 8) gauge wires take the power from the 20 amp breaker in the sub panel to three separate outbuildings via underground PVC conduit. If those supply wires became compromised (e.g. if a gopher chewed through the PVC and wire insulation), that would also trip the breaker, wouldn't it? I'm hoping I don't have to dig up 150 feet of conduit to find the problem :D

I don,t understand how you can disconnect the breaker from its branch and still have it connected to something in another room? I mean actually disconnect the wire at the breaker NOT just turn something off.
It,s also rather unusual to have 3 seperate buildings supplied from the same breaker. You would need a splitter somewhere (or the illegal 3 wires under the same screw on a breaker)
If you are concerned about the integrity of the underground run Disconnect the loads at the first junction box in the barn where the wire comes out of the ground downstream from the breaker Then you will ONLY have the undeground wire connected to the breaker . If it then trips the problem is in the ground. You don,t dig you attach a rope to the wire and pull it out of the conduit. Repair or replace and pull it back in with the rope.

Also be sure to do this disconnecting with the power turned OFF to your breaker panel at it,s supply.
If this basic troubleshooting is this confused perhaps you should call a licensed electrician.

I'm obviously not communicating what I'm doing properly.

Right now, the breaker is connected to the sub panel, AND the supply lines that feed the individual buildings. In front of each building is a PVC junction box underground. Connected to the main supply lines at each point is 12/2 direct burial Romex (in PVC conduit) that goes into a metal junction box in each building. From that metal box, additional 12/2 Romex goes to the various outlets and light switches.

At this point, I'm testing to see if any of the "branches" from the metal junction box has a short circuit in it that would trip the breaker. I disconnected one of those last night, but the problem remained. I plan to disconnect each additional "branch" one at a time to see if the problem gets resolved.

If disconnecting each "branch" does not solve the problem, then the only remaining possibility is that the short is in the main supply wires.

One of my initial questions was whether or not there was a way to test for a short circuit in one of the branches without disconnecting them one at a time. It seems that there isn't, but no one had addressed that directly.

Hope that clears up the present situation.