Barkeeper's Friend first. Oven cleaner if the first doesn't work, but only on the stainess. Also, high heat will discolor stainless without there being any deposits there. Learn the difference as that is a permanent change to the stainless that nothing will change. It's a badge of honor that the stove isn't merely a decoration.

I appreciate your suggestions Barkeeper's Friend contains oxalic acid. Would asolution of that chemical work? The fumes from oven cleaners used to be pretty horrific.

Bar Keepers Friend is safe and great for just about anything!!! I use to get soap scum off glass shower doors, go back a wash with the platinum liquid dawn and dry with microfiber cloth. They will look brand new. I use on caked on casserole-dishes, my cephalon cookware amazing on glass stove stop ranges and very inexpensive :)