Grease
If you do a lot of cooking with grease, you will often be left with a mess in front of your stove on the floor. For a no wax clean linoleum floor use one cup of vinegar, one quarter cup of baking soda, one table spoon of dish soap, and two gallons of hot tap water. You can use this mixture with a mop or with a scrub brush to get up the grease in front of the stove on the floor.
Juice
Sticky juice spills that will crystallize on your linoleum between cabinets and the fridge will disappear with just plain hot water and a rag. Using hot water to loosen the dirt the juice will come right up.
Pet hair
If you have pet hair that accumulates under baseboards and around your trim work you can use a rag that is wet with vinegar to catch the hair while you are deodorizing at the same time.
Ammonia
For a clean shine that removes dirt on your linoleum floor, use four cups of ammonia with a gallon of hot water to clean your no wax floors.
Cat litter trays
If your cat litter makes a dull area around the area where the litter box is kept you can clean this area with straight vinegar to cut germs and to give your linoleum the boost it needs for a fresh looking shine.
Crayon
Crayons on the linoleum floor can be scraped off with your fingernail, a soft rag with hot water (to actually melt away the crayon mark) or you can use a soft rag and the heel of your foot to scrape away the crayon marks.
Latex paints
Latex paints that are spilled on the linoleum and then dried will often peel right off the linoleum. If it is tiny splatters you can use vinegar straight on the painted areas and then simply rinse with hot water. Never use a scratchy pad or other type of scratchy tool or you will leave your linoleum looking dull.