Televisions have a conductive coating on the sides of the picture tube. A grounding wire actually penetrates through the glass tube to the conductive coating on the inside. This is to make a complete circuit for electrons to flow in. Electrons are shot from the electron gun at the phosphor screen in front, and they have to have a place to go after they've hit the phosphor.
I can imagine that good shielding of an oscilloscope (or even a television) can help picture stability, in addition.
You can't really do a good job discharging insulators, since charges are not mobile inside them or on their surfaces. About the best you can do is to rub their surfaces gently with a grounded conductor. I used to have a really cheap plastic chair which would get all charged up with static electricity every time I got out of it. Just passing my hand over the charged areas was enough to (mostly) discharge it, but I didn't always get all the charge off all the spots. Eventually, over time, charge on the surface of an insulator will leak into the air, but charge embedded inside an insulator can stay a long, long time.
Another way to reduce static charge is to use a little anti-static gun. These have piezo-electric crystals which generate voltages when you squeeze the trigger. One side is connected to a sharp metal point, at which very large electric fields are created. Ions are generated there. As they spray out, they are attracted to oppositely charged regions and repelled by like charges, so they reduce the static charges. If a balloon is stuck to the wall by static charge, you can shoot it down with an anti-static gun.
(published on 10/22/2007)