I've fixed all of the lights on this car with the exception of the running lights. The flashers, turn signal and brake lights all work, but when I turn on the headlights, the running lights in the front and rear don't work at all. Does anyone know where the most likely place is to check to repair this issue?

Headlight switch/bulkhead connector.

I've been through the bulkhead connector, cleaned the contacts with cleaner and it seems fine. The headlight switch does seem to be an original. The only lights that don't work as they are supposed to are the running lights (tail lights and front turn signal lights) when the headlight switch is turned on partially or fully.

Any other thoughts as to where to look for this issue?

Have you verified voltage at the light sockets? I'd start there and work back to the bulkhead.

I have not, since the brake lights work in the rear along with the rear turn signal lights & flashers. Everything works but the running lights when the headlights are turned on at the headlight switch.

Brake lights, turn signals and flashers are a completely different ciruit from the running lights.

That's kind of what I thought, which is why I'm asking about common places to troubleshoot this issue. All of the other lights are now working, except for the running lights.

As an FYI, I connected the battery, turned on the headlights and checked each fuse with a test light. The only fuse that didn't have power at each end was the fuse circled. In addition, the cigarette lighter was indeed in the car. I removed the three screws that held it in-place and found the double-wire connector attached to the back of the cigarette lighter. I removed the connector and checked it with the test light and no voltage was detected there either. Thoughts on how to proceed with my non-working running lights issue?

Did you have power at either end of that fuse? if you did but only have power at one end replace the fuse.

I have power on the outer portion of the fuse circled in "red". The side circled in "yellow" had nothing. I replaced the fuse, but it blew. I can't recall if it blew immediately or after turning on the headlight switch. I think I'm going to replace the headlight switch for starters, as it seems to be the original switch and it makes a crunchy sound when turning the headlight switch knob, in addition to seeing the white circular portion of the switch walk back and forth when observing the switch in the dash from below.

that white circular part is the rheostat that brightens and dims the dash lights.Replacing the switch sounds like a good idea.

In addition, the dash lights don't work either, so there's another reason to replace that pesky headlight switch... Also, had to replace the brake light switch on that car to restore the brake lights, as the brake light switch was seized and blowing it's associated fuse...

There is a black wire that runs from the bulkhead connector (I think) and goes past the battery and just as it gets to the other side of the core support in front of the battery,has a soldered joint that splits a ground to electrical components up front. Peel the tape back and check that splitter. I had a similar problem and found the joint was corroded.

Thanks for the update. I'll pass that information along to the car owner, since he didn't seem to concerned about the issue, as he doesn't plan to drive the car at night, as of now.