The first thing I'd do is check what the voltage is at the outlets.

It may be a loose connection in the receptacle, possibly a backstabbed white wire or feed-in/feed-out using both screws instead of a pigtail. Any surface that is too hot to touch for a few seconds should be investigated if you are unsure of the cause of the heat.

Do you have aluminum wiring?

Ditto on the voltage check at several different outlets in different parts of the house. If some are higher than others, could be a loose neutral.
Story too long to post here in full detail, but MANY years ago, about 6:30 pm on Christmas eve, got call about failed water heater in a mobile home. Heater was under the cabinets in the kitchen, non-replaceable element burned out, 120 volts. I did have a new heater that would fit the space, 240 volts. Installed the heater, changed connections in the electrical panel and rushed home to the big dinner awaiting. Some days later the customer asked me if I changed any other wiring. Not to my knowledge. Turns out that one of the counter top receptacles was on the same circuit with the water heater and now was fed with 240 volts. As the customer described the results when a mixer was plugged in, it was difficult to keep a straight face. He told me that it was really good if they were running late in the morning as the percolator would make coffee very fast if plugged into that receptacle. The problem was corrected at no additional charge.

I thought we had a voltage tester but I guess not, so as soon as I get to the hardware store I'll test the outlets. In the meantime, how would I tell if the wiring is aluminum?

Do you know when the house was built? Aluminum branch circuit wiring was installed in some homes from approx. 1965 to 1973. It is easy to identify, just peek into one of the problem boxes with a flashlight and see if the bare part of the wire is copper colored or aluminum colored. Shut the power off first of course.

This post was edited by joefixit2 on Sat, Aug 2, 14 at 23:46

Why the seeming focus on Aluminum wiring when the issue is apparently high voltage?

Just asking.

"Why the seeming focus on Aluminum wiring when the issue is apparently high voltage?"

Hot receptacle outlets and melted switches point to connection issues which are common with aluminum branch circuit wiring. Connection issues can cause voltage variances.

This post was edited by joefixit2 on Sun, Aug 3, 14 at 11:06

But poor connections would cause low voltage. Only at the service neutral would a poor connection possibly cause higher than normal voltages.
Overheating from poor connections occurs at the connection, not within the appliance unless the problem connection is within the appliance.

What about a MWBC? I have seen 2 fires from neutrals of a MWBC backstabbed. Both started in TV sets that were using the MWBC when the backstabs failed. The OP mentioned hot receptacles, a melted switch and power dropping in and out at some receptacles, which is why I think there are connection problems at the devices.

Pretty tough to diagnose anyway from where we sit, a local pro should probably be called in at this point. Could be a dangerous situation.

We live in a townhome built about about 12 years ago. I took the plates off of a couple of outlets and most of the wires are coated, so it was hard to tell, but I caught a glimpse of what looks like copper wire.

After reading about what a MWBC is and talking with a neighbor, I'm going to call an electrician. If I understand correctly, since a MWBC is a sort of a chain, if there is a problem with one connection then it can cause problems with all of the other related outlets, etc. which makes a lot of sense. It also makes sense that I would not be able to find the problem on my own, since it's not just a matter of switching something out!

Thanks for the help!