Who should write your recommendation letter?

US universities generally require two teacher recommendations and one counselor recommendation.

For the teacher recommendations, you should ask NUSHS teachers who have taught you, preferably in Year 5 or Year 6. If you're taking NUS classes and want to get recs from the NUS professor, I will still recommend that you ask NUSHS teachers because your relationship with them is probably closer and thus they will write more meaningful recs. I took NUS courses but I hardly interacted with my NUS professors. However, I was always in contact with my NUSHS teachers through classes, consultations, extracurricular activities.

Many NUSHS students conduct research at external organizations like A*Star research institutions or at NUS. If you have a close relationship with your research mentor, you can ask him to write a supplementary recommendation letter. This recommendation will not be submitted through the Common App system and it does not need to be confidential. The easiest way is, your prof writes the letter with his organization's emblem and you send it to all the schools via fax or snail mail.

From Cal Tech's website: http://www.admissions.caltech.edu/faqs#recommendations
Do you accept supplementary recommendation letters?
Yes. If you have conducted research, we strongly encourage you to submit a letter of recommendation from your mentor. You also may submit other recommendation letters if they add information to your application that we would have not otherwise known.

Can recommenders send the recommendations via email?
Recommendation letters should be submitted either via the online Common Application [Fiona: Supp recs can't be sent via Common App] or via regular mail. We do not accept letters of recommendation via e-mail.