Since that black wire was melted, and the breaker didn?t trip, either the breaker you changed was bad, the breaker rating is too high for the wire size, or the connection was loose or corroded. ( that would be my guess) Typically in 120 ac service, the black wire is normally hot, the white wire is the neutral, and the green or bare copper wire is the safety ground. The green wire and white wire normally go to the same ground location in your panel, but the serve two different purposes. If you tested and showed no voltage from hot (black) to the neutral wire (white), then you don?t have a complete electrical path back to the panel on the neutral, which is why someone already suggested to check for tightness on the neutrals at the panel. However, if it?s been loose for awhile, it can cause corrosion at the connection and just tightening it won?t help. You would have to pull it and clean it. Usually, there are several receptacles wire in series. If you test all the receptacles and NONE show voltage from hot to neutral it?s most likely not making connection at the end of the run, the panel. Or in could be the last receptacle in the series could have a bad neutral connection. Knowing which order the receptacles are wired is probably impossible to know.
One easy thing to do is kill the power to the panel and relocate the neutral wires to different connection holes at the panel. Most panels usually have spare holes on the neutral buss bar. If not, I would pull each one, inspect it and clean if necessary, then re-install. If that doesn?t work, you are back to a bad receptacle connection on the neutrals, or a broken wire which I think highly unlikely. Some cheap receptacles are made when you just stick the wires into the back of the receptacle and a thin piece of metal inside locks the wire into place. These styles will have a little slit right below the hole where the wire goes in. That slit is to stick a small screw driver or equivalent into that slot to release the wire from the receptacle. The problem with these is that small locking blade only contacts a very small portion of the wire surface and it makes them really prone to overheating or poor connections. I personally never use this style because I consider them cheaply made and dangerous. Anyway, that?s where I would start. (What Gary said, I was posting as Gary was writing I guess.)