I am not sure if you can *remove* the spots, but some things will shine up the SS so that you don't see them.

By far the very very best SS cleanser and polish I have found is called "signature polish" it actually cleans the SS and leaves a nice finish on it that resists fingerprint but is not greasy. It looks like soft scrub, but not really gritty. I got a tiny sample of it when I got a Wolf stove. . So you will likely have to go online or to a specialty store to get it.
I have also tried "sheila shine", which is really more like a polish, but leaves kinda of a greasy shine. It is a spray on polish. I was able to find sheila shine at lowes.

I have also used Bar keepers friend on stainless steel to remove some spots...(or rust).
Hope that helps! (and that I am not too late)

Stainless steel is an alloy of iron which contains more than 10% chromium. Before you use your Stainless Cookware for the first time, wash it thoroughly in hot soapy water with a sponge or dishcloth.
Immerse your in warm water. Use a fine powder cleanser such as "Bar Keepers Friend" "Wenol", "Twinkle", "Cameo" or "Bon Ami" and form into a paste. Apply the paste using a soft cloth. Rub the paste in a circular motion spiraling from the center outward. Wash again in hot soapy water, and dry immediately. DO NOT USE OVEN CLEANERS OR CLEANSERS WITH CHLORINE BLEACH OR ALCOHOL. DO NOT USE STEEL WOOL*.
Use this, its really works.

If the stainless with a spot happens to have a brushed finish you may be able to remove it with some auto refinishing emery product.