How to repair shallow scratches in brushed stainless steel
I scratched my brand new dishwasher. It was brand new, and so tragic...

I opened an overhead cabinet and a wine glass fell out. I instinctively caught (a piece of) it between my leg and the dishwasher, and the shard of glass scratched it :-(. 

So, you can't remove big gouges without some sort of fancy kit, but you can take care of minor scratches with common household items. 

To start, here is the scratch (vertical mark) and the gouge (the horizontal mark):

You will need a size 0000 extra fine steel wool scrubber, and a little vegetable oil. 

1. Using your finger, put a generous dab of vegetable oil on the scratch. 

2. VERY, VERY gently, start wiping with the steel wool. Always go with the grain. 

With most appliances, the stainless look is simply a coating (they are not stainless all the way through). Start by wiping as gently as possible, then slowly add a little more pressure bit by bit until the scratch starts to disappear. You do not want to go through the coating.

This is why it doesn't work on deep gouges, only on small scratches. 

Here it is after I gently scrubbed (using the base of a tea spoon for reference.)

 4. Clean off the oil with a hot cloth, then wipe with some stainless steel polish. 

Here it is after I washed off the oil and used a little stainless steel polish:

This process was able to get rid of the scratches, and also lessen the gouge a little. 

On pots or pans, kettles, you can scrub harder, since they are stainless all the way through (just always go in the direction of the grain).

Before and after!