I am trying to get a Singapore multiple entry visa. My question is: can I work in Singapore on this visa base legally? After how many days do I need to come back to my home-country?
You won't be able to work on the visa.
What you need to get in order to legally work in Singapore is some sort of work pass or permit, which usually your employer will have to get for you. However, there is no issue with looking for a job while on a visa. Once you get a job, your employer can apply for a work permit or pass on your behalf, which will give you a long term pass to stay in the country.
Until then, you will most probably not be able to stay any longer than 30 days at a time in Singapore.
DO NOT overstay or work illegally, as this will most certainly get you into trouble, and other than being potentially fined, caned and/or imprisoned, you will also be very unlikely to be granted a visa again in the future.