Hi Amanda!
My favorite "harness and cable" sighting is right at the beginning of Season's Greedings when Superman floats down to save Danielle from the snowballs. If you look on his right hip, you can catch a quick glimpse where the cable is attached. You can also see the one on his left hip too if you look for it, but it's a very quick shot either way.
But when he floats upwards saying "Merry Christmas", you can see them both clearly.
I know there are many, many more, but that's the only one I can think of off the top of my head. Believe me, it's not only kids that miss this stuff--I was an adult when LnC was on and I didn't see many goofs back then. I too was so busy watching Dean that I didn't see much else!