Avoid Chlorine Bleach
Try to clean of as much rust as possible from the surface that has rust on it. Make sure that you do not use chlorine based bleaches as they often make things worse. In case chlorine bleach is used, it is expected to cause discoloration to the spot where the stain is.
Image Courtesy: amazon.com
Liquid Soap and Water
The best way to go for gentle rust stains is to get some liquid dish washing soap, preferably around half a tea spoon and mix it with a cup of warm water. Soak the stained area with the solution and let it sit for around five minutes. Afterwards, dab the spot with a paper towel or a plain white cloth. Do not rub it as the stain can get worse with that and the surface of certain cloths can get damaged.
Afterwards, rinse the stained area with cold water to rid of all the rust. Make sure that the rinsing is thorough as otherwise some of the solution may be left in the cloth or carpet.
Image Courtesy: paintingwithwatercolors.com
Lemon Juice and Salt
You can also use the lemon juice and salt method. Pour plenty of lemon juice on the surface where the rust stain is. Make sure that it is soaked. Afterwards, put some salt on it and let the two sit on place for a fair bit of time. You may add some more juice with time to avoid it from becoming dry.
Afterwards, preferably 24 hours later, dab the area with a soft cloth. Wash the area with cold water. If the stains are not completely removed, repeat the process once again and the surface should be all cleaned up without damaging the surface.
Image Courtesy: motherwouldknow.com