Running Lights Trouble

It sounds like you have a loose wire. A fuse would cause all the lights to turn off. I assume that you are talking about a trailer. If so, then having the front lights on shows that the TV plug is working. It is not likely to be the ground that is loose because the brake lights usually use the same ground as the running lights, if it was a ground then the brake lights would not work. It has to be the + wire that is bad. Most trailer lights are daisy chained, have wires that go from one light to the next to the next etc. When a wire gets lose everything after the loose spot turns off. In my trailer the chain starts at the front right, across the front to the left, down the left side, then across the back to the right side. If you follow that pattern I would remove the first light that is out from the trailer to get at the wires behind it. Look to see if there is a bad connection, and check the voltage at the wires. If everything looks good I would check the last light in the chain that is still on. If connections are tight at both lights, then probably a mouse has chewed the wires between.