Which State should I register my business?
Being a non-U.S. resident, if you do not plan on having a physical presence in any one particular state (such as an office, employees, etc.), you could register in any state, and you may even list your company in Singapore as partial or whole owner of the U.S. entity.
We have a research article titled 'U.S. Companies for Foreigners', which offers a side-by-side comparison of the top recommended states, and guides you through the online application process.
To accept payments, you would need to open a merchant account, which can actually be done without a U.S. bank account. We are able to put you in touch with several providers of merchant services who will send you their quotes for rates and terms. This free option is located in the "Bonus" section of any of our entity registration applications.
Regarding U.S. bank accounts, this is a challenging problem if you are unable to physically travel to the states and apply to a U.S. bank in person. We are trying to resolve this issue and are hoping to have a sure-fire solution in the near future. For now, please refer to our 'Banking in the U.S.' article to see what options are available.