Maintaining linoleum based floor is not really a hectic job to do. As such linoleum is quite a stable product and is durable so you need not fear or worry your head too much with its maintenance. But then there are some specific products that can lead to damage of linoleum based floors actually. You must make it a point and avoid those things. Like, you must keep away all those products which are by nature alkaline. Linoleum is quite vulnerable to these.
There are solvents that can also cause a significant damage to the linoleum based floors actually. These solvents can render your linoleum quite soft and this is a condition which is really not easy for you to overcome. So avoid these types of solvents. Again at the same time you have to keep in mind that you are not using real hot water in order to scrub the linoleum floor. There would be scratch marks and abrasion marks that you will not be in a position to get rid off.
For best results it is a good idea to use an all purpose cleaner for your linoleum flooring. You can dissolve this cleaner in warm or luke warm water and then you can choose to scrub your linoleum with that. Two coats of self polishing solution can be really ideal for you as well. It would also give a very shiny look to your floor. Two coats is just the right amount that I can suggest you, but if you are in real hurry and want to see the results of quick action then perhaps one coat would do you justice. That would give you enough time as well to render the floor a fresh and good look.
Well it is completely upon you to decide on what is the degree of cleaning that your linoleum would require. If you think its high time that you need to give it a heavy duty cleaning session then you need to strip it off the wax surface first actually. For this you need a very effective wax stripper and are very careful to make note of the specifications that are required for this. In a bucket of water you can make a solution containing vinegar that is the food grade one and water of course. The solution should contain 50% of vinegar and rest 50% water. So with this solution you can choose to clean up the things properly.
However if you have no wish of cleaning your floor but want the shine to come back for a temporary period then you better get hold of vinegar solution. It is really effective in providing you with that shine actually for at least some time.
To get rid of spots over the floor you should get hold of steel wool, which would solve your purpose greatly. Damp clothes are also good so you can jolly well go for that and let it dry after some time.