[Al Skierkiewicz]

Phil,
Great pictures, now we are talking. The output waveforms from the wall transformer are normal outputs of a full wave rectifier with very little filtering or too great a load. Not ususally a problem for a 78XX when the lowest part of the waveform is still 2 volts greater than the output. Look close at this waveform and you will a lot of fuzz on the ripple. Since you have a dual trace scope, if you were to put your other probe on the output of the 7805 you would see that the two occur at the exact same time. If you were to delay trigger on the noise you are going to find it is a fairly high frequency, <1MHz. Since it is locked to the line, you have local interference from a line operated device. Most likely a dimmer. If you have a dimmer in your house, try turning it to a different brightness and the period of the noise will change. This could be a fan control, a touch on/off lamp or a wall dimmer. (X10 dimmers are the same.) This could also be caused by a bad switching power supply but that is pretty rare.
The fact that the waveform gets worse when you touch the backplate just indicates that there is a strong field picked up by your body and capacitively coupled to the backplate.
In rare cases, this could be caused by a bad insulator on the power line. Take an AM radio and tune it to a station that is not close to your house. See if you hear hum in the audio and then use the radio to point you to the offending device either inside or out. Am radios use a bar antenna and the greatest sensitivity is along the axis of the bar. By turning the radio you can get an idea of where the culprit actually is. If it turns out to be a bad insulator, you need to contact the power company and tell them you have interference from their equipment. They are required to make the repairs, although not in a timely fashion.
You may be able to make things happy by grounding the back plate to the ground in your house. Run a wire from the chassis of the scope to the backplate and see if that makes things better or worse.
Once you get this part cleared up, we can talk about the what the waveforms on the power transformer are telling you.