Depending on the surface finish of your stainless steel, abrasive cleaners can cause scratching. If you have a dull finish, it probably will not show scratching as much as mirror or highly polished finishes tend to. Usually, stainless steel won't need to be scrubbed. When in doubt, test in a hidden spot. It's also a good idea to work from the least risky type of cleaning to the heavy duty stuff. Start with water before you move to commercial cleaners.
Do not forget to rinse.
Gritty or dirty water can leave a residue on your finish. It can also stain or pit the surface of your stainless steel. Be sure to rinse completely. Similarly, residue from cleaning solutions left on a stainless steel surface can stain or damage the finish. Rinsing is a key component of cleaning stainless steel.
Do not use cleaners containing chlorine.
While it may be second nature to bleach everything, stainless steel and chlorine do not mix. Stay away from the bleach when you clean stainless steel. Be aware that bleach can be included in different types of cleaners. If you accidentally get a cleaner on your stainless steel you'll need to rinse it off and quickly.
Do not use steel wool or steel brushes.
These products leave little particles in the surface of the steel and inevitably these particles begin rusting and staining the surface of the steel. They also can excessively scratch the surface of your stainless steel. Stay completely away from steel wool and steel brushes.
Do not assume it's the cleaner.
If you do have some spotting or staining, and you've followed all of the rules, it may not be the stainless steel cleaner. Water, especially hard water, can leave spotting and staining on stainless steel surfaces. Towel dry after rinsing can end the problem. Sometimes, even water is an enemy to stainless steel.