Scary stuff, 120 volts AC, when you're connecting it to something this sensitive. Very easy to let out the magic smoke. By the way, this isn't just dual trace, but dual beam, which is even better. I think you can put dual timebases in it, or do something like text and signal onsceen at the same time.
Anyway, yes, use the x10 probe for safety. Then you'll only be seeing 12 volts AC on the output of that. The probes are designed to not distort the waveform, just sample a small part of it.
If you use an isolation transformer, I'd put it in the power line to the scope. You don't want it in the signal you're trying to look at because you don't want it doing anything to the waveform. You may not need one.
One old safety tip from an old Radio Amateur's Handbook with dealing with circuits like this with lots of unknowns is to keep one hand behind your back. Stand on a rubber mat and/or wear rubber soled shoes at least until you've figured a few things out. No voltage will hurt you if you're only making contact with one side of it at a time, which is why birds can sit on power lines.
I can't tell from the picture but it looks like the BNC input jacks on the front have insulating washers under them, so they aren't connected to the chassis. I'd get a trace going across the screen by fiddling with the trigger level, set the vertical sensitivity to 20 volts/div, put on the x10 probe and touch the end of it to what you think is a hot wire. You should see something change on the screen. If you don't it may be because the scope ground isn't connected to the signal ground. Before you connect it try to make sure you aren't going to fry anything by doing it. There may be some voltage between the two (check with an AC meter or cheap neon test lamp), but no current should flow. If you see a voltage difference, try something like a small 120 volt light bulb across the two. It shouldn't light. There may be some voltage which will go away when you connect them, but if there's current flowing you'll fry something. If you get current you'll need the isolation transformer. Plug the scope into it and start over.
One thing to bear in mind is that what you'll see on the scope screen is most easily considered as peak to peak voltage, that is, the distance from the bottom to the top of the waveform. When most people talk about 120 volts AC, they're usually talking about RMS (root mean square) voltage. Peak to peak is about 2.8 times RMS. So 120 VAC RMS should look like 120 * 2.8 = 336 divided by 10 (the probe) = 33.6 which at 20 volts/div would be about 1.7 divisions. Once everything is under control you can turn up the sensitivity to see the waveform better. If you're only seeing only half this much and you didn't ground the scope, that could be why. At 60 HZ you should see one full cycle in 1/60 = 16.67 milliseconds, so 5 milliseconds/div (the 5 over the m) setting should show something useful.
Alan
RMS