The good thing about linoleum floors is that they are easy to clean and require comparatively low maintenance. Paradoxically, that causes most users to use linoleum floors so roughly that they often take far more than their normal share of wear, tear, and stains until their shine is lost. Being a natural material that is derived from powdered linseed, linoleum does not react favorably to water and alkaline solutions, so be careful to use little water and no bleach when removing stains from linoleum.
A very good anti-stain measure is to polish the floor regularly to prevent the accumulation of dirt, grime, and assorted stains. Removing stains from linoleum can often be a tricky business because you may have to use water, which will make your floor vulnerable to moisture damage. Moisture can decrease the natural antibacterial powers of linoleum, which make it mold and mildew-resistant. Therefore, make sure you use only reputable linoleum cleaners and follow the manufacturer's instructions to the letter.
Since most stains on linoleum can be removed using the same cleaning techniques, it would be more convenient to discuss all-purpose cleaning methods for various stains rather than talk about specific solutions for particular stains.
For relatively fresh stains, sweep or vacuum the linoleum to make sure that no dirt particles remain. Next, dilute the required amount of your preferred linoleum floor cleaner in a bucket of water (about one gallon is a standard measure) and, using a clean rag, wipe the floor gently. Should you find that the rag or the water is getting too dirty, remix the solution and replace the rag. Once the stain has come off to your satisfaction, air dry the linoleum.
For more stubborn stains, first free the floor of all excess dirt and grime using a vacuum cleaner. Next, dilute a mild detergent in water and mop the stained area or areas lightly with a clean rag soaked in the solution. In order for the detergent to work on the stain, you must let it stand for some time, though be careful not to wait too long since water can damage your linoleum. After about five minutes, wipe off the detergent with another rag dampened with plain water and let the area air dry.
Alternatively, once you have vacuumed your floor, remove all traces of wax from it using a commercial wax stripper available at hardware stores. These are usually accompanied by manufacturer's instructions and ought to remove the wax thoroughly. Once all the wax is gone, mix white vinegar and lukewarm water in equal proportions in a spray bottle. Spray the floor with the vinegar mixture, and then wipe it off with a damp cloth. This technique should take care of all kinds of stains. Once you have dried the floor thoroughly, re-wax and polish it according to the manufacturer's instructions.
Remember, with proper care, linoleum can look good for years and years, but you must assume that responsibility. Do not allow dirt and grime to accumulate on the floor, because larger particles tend to abrade the surface, thus making it more prone to stains and scratches. Finally, avoid using strong chemicals, which may destroy the natural properties of linoleum.