What is the best way to clean a linoleum floor?

Vacuum and damp mop for most cleaning. Maybe a little detergent and then a clean water rinse if needed.

I use dish soap or castile soap.

Vinegar, dish soap and water!

Use the above cleaners but use a microfibre cloth on a Swiffer mop head! No need for an expensive cloth, dollar store variety works wonderfully. When done, pop off the cloth and wash in the washing machine, air dry!!

I'm Canadian, and one of our local hardware stores sells a microfibre twist mop. This thing is fabulous. It unfortunately has one cleaning pad. The mop head end butterflies and the center part of the pads is fixed. It cleans my floor extremely well, when I remember to wash it. It also works well to wash windows and especially walls, without climbing stools.

Is there something that you can't clean out of the tiles that is especially bothering you? For me it was nicotine stains on a grey floor. The previous home owner of my house were heavy smokers. After years of full strength floor strippers and green scubbie pads to remove the nicotine, I think the colur has been leached out. I then put multiple coats of acrylic floor wax. Using my mom's old floor polisher, give the floor a nice buff. Nothing looks nicer than a shiny clean floor.

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You are going to want to get a floor scrubbing machine. I have a very old one but it works so well... I love it! Something like this....

https://www.amazon.com/Gloss-Floor-Scrubber-Polisher-B200752/dp/B00063KZBA/ref=sr_1_5?crid=P5CV392AFOH2&keywords=floor+scrubber&qid=1554815451&s=gateway&sprefix=floor+scrub%2Caps%2C154&sr=8-5

You will get a great scrub and you don't have to scrub on your hands and knees! It works great on linoleum floors. I have heard great things about the steamers too but I grew up with linoleum floors and this is the type of thing that we used.