Swimming without goggles in Singapore swimming pools is generally considered safe. As swimming pools are chlorinated to safe levels, you are more likely to experience some burning and irritation due to the cleaning agents in the water.
Such redness and eye irritation is usually transient and temporary. Lubricating eye drops will soothe the eyes and the eyes should return to normal very soon.
Outside of Singapore or in polluted waters, you may be at risk of contrating a variety of micro-organisms including bacteria, viruses, and protozoa including acanthamoeba that would put you at risk of an eye infection.
This risk is higher if you are using contact lenses at the time. Avoid prolonged exposure of your eyes or mucous membranes in such a situation, or better still, don't swim in such a contaminated body of water!