As a web designer, online visibility is incredibly important when it comes to branding globally. Depending on your business, a lot of your global business goals can be achieved by setting up an online business.

You should register domains in the countries you'd like to market to, such as the US, NZ, and SIngapore that you mentioned. Also establish a Business Plan and central brand that includes all of these markets.

Try to avoid getting caught in a re-branding process down the road. If your brand needs tweaking in order to achieve scope, do so.

I would start by registering the business in the country you mainly do business in, and provide shipping of products/services to all of your other markets if necessary to get started. Then as you establish a foothold in each territory, look into the benefits of setting up locally.