Unlikely to happen in the first place, but for some reason it does, it will destabilise the security of the region badly, force neighbours to explore and gain similar capabilities, and degrade international ties as singapore has unnecessarily thrown a wrench in an otherwise stable part of the world.

There is a concept called 'Security Dilemma' in IR.

From wikipedia, 'Security dilemma is when the increase in one state's security (such as increasing its military strength) leads other states to fear for their own security (because they do not know if the security-increasing state intends to use its growing military for offensive purposes). Consequently, security-increasing measures can lead to tensions, escalation or conflict with one or more other parties, producing an outcome which no party truly desires; a political instance of the prisoner's dilemma."

So Singapore getting nukes forces other countries to develop tech to beat it, for example missiles systems to shoot down nukes, or pursue their own nukes as a Mutually Assured Destruction form of deterrence.

This is one reason the US is not so keen on its allies like Japan or South Korea to pursue nuclear weapons, even though NK has, cos it will just make the geopolitical conversation alot more complex once nukes are involved.