There are 19 members from this family listed in the 2009 Singapore Flora checklist. Among them, only 4 were given the naturalised status, meaning they can be found growing without human intervention in the wild. The 4 were Solanum americanum, Solanum lasiocarpum, Solanum sarmentosum, Solanum torvum. Based on my observation over the years, the mentioned status may not be totally correct.
(1) Solanum americanum: The species found in Singapore is most likely to be Solanum nodiflorum. I did a rather thorough check on this plant when I grew it a few years back. Indeed, this plant is naturalised though it is relatively rare.
(2) Solanum lasiocarpum: This plant is commonly known as Indian nightshade and is found in the wild in South Asia. I had not encountered this plant in the wild here. The naturalised status is probably incorrect.
(3) Solanum sarmentosum: According to the Plant List website, this name is a synonym of Solanum lanceifolium (lanceleaf nightshade). It is a vine and information on it is scanty. Some pictures are available in the Plants of Saint Lucia website. Again, I doubt this plant is naturalised here.
(4) Solanum torvum: There is no doubt that this relatively common shrub is naturalised here.
In addition, there was one member that was naturalised here but was not assign this status --- Physalis minima (Bladder Cherry). In summary, the 3 species that were naturalised in Singapore at this point in time should be Physalis minima, Solanum nodiflorum and Solanum torvum. The assignment of naturalised status to a plant can be a tricky business. Plants listed in the Flora checklist were collated from previous publications dated as far back as 1990. It might be true that some of these plants were naturalised then but had gone extinct over the years.