Most of the advice out there is written by people trying to sell you something. Here is the un-sponsored version, from someone who chose local and partly regrets it.
The case for local
It is cheaper. It is closer to family. NUS and SMU have genuinely world-class programmes. The graduate employment rates are excellent, especially for finance, law, and medicine. If the programme you want is strong here, there is nothing wrong with staying.
The case for overseas
Three things, in order of importance.
One, the life experience. Living alone in a different country at 19 is formative in ways that are hard to articulate. I underestimated this.
Two, the network. The person sitting next to you in tutorial who becomes a lifelong friend is more likely to be international, which matters if the career you want is international.
Three, pedagogy. Not always, but often, overseas undergraduate teaching is more discussion-based and more student-driven. This suits some students much better than the lecture format.
The thing nobody tells you
You can come home. People forget this. Studying overseas for three or four years and then working in Singapore is a perfectly normal thing to do. The fear that you will "get stuck" abroad is usually overstated.
What I wish I had known
The cost of overseas is real but smaller than it looks once scholarships and financial aid are factored in. Some of the top UK and US unis offer need-based aid to international students. It is worth a proper look before ruling it out.