What's more likely an objects whose mass is 0 or object whose weight is 0

It is more likely for an object to have a weight of zero because the weight is the measure of the gravitational force on an object. It is possible to weigh nearly 0 in outer space. Matter, however, is the measure of matter in an object. More or less it can get down to how many atoms in the object. There would be no object if the mass was 0.