Which brokerage would you recommend for Singapore and USA stocks and why?

Best to use separate brokers for both markets. The ones that allow you to trade SGX stocks are kinda expensive commission wise for US market. For local, there's not much difference between the brokers so you may wanna see how your broker service you. I'm using Maybank KE for local as my broker is responsive, and I'm also on their monthly investment programme to help build my income floor portfolio.

US wise, I'm using TD Ameritrade, mainly for options. Great and robust platform, with none of the misc fees you see on other platforms *ahem Interactive Brokers. Charting tool is great, straightforward interface. They have an office here as well, so easy to get hold of them if needed to.

Hope this helps.