When we bought our house I went through all the linoleum areas and used a scrubby with comet. I used a Scum Buster too when my arms got tired...I had forgotten I had it or would have used it the whole time, I have tons of extra batteries charged.
After cleaning an area I rinsed and rinsed. Then once the whole room was done I checked again for any residue and rinsed again if needed. This took the floor back to nearly the basic flooring, it had a dull finish at this point.
I bought floor wax at Lowes, it is by the carpet and it says it's for linoeum and other finished floors. It was about $15-$20 and the bottle wasn't as big as a gallon. I used a regular new sponge mop and waxed the floors, let it dry completely, then put on a second coat. I use the wax every few months to keep the floor looking brand new. It looks amazing all the time. Nothing stains it or makes it look bad.
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Now I use my steam mop for all cleaning and it melts all goo. It leaves the floor really clean and shiny.