How do I remove thick, stubborn soap scum from my shower door?

Mixture of Vinegar and Dish soap.50/50 Ratio.

Barkeepers Friend powdered nonabrasive scrub from the grocery store, a plastic scrub bud and a little elbow grease. After you get it clean apply RainX for car windshields to keep the scum from building back up for a fairly long time.

Use a nylon scrubber dipped in a 50/50 mix of white vinegar and water. It may take a few tries, especially along the edges.

If you can get it, CLR is great for this as well as many other hard to clean things.

Spray the glass heavily with a cleaner and use a razor blade to scrape it off. I have done this a few times to mine and they look great. Just keep the glass wet with Windex or some other cleaner. Wear gloves so you don't get cut. It takes a little time but it does work, than use rainx.