What’s the best budget route to travel via flight from Singapore to Austin, Texas?

It is hard to say, "Best" as there are sooo many factors affecting your flight. Are you flexible with dates? Flexible with timing? How many stops are you willing to make? Is the reputation of the airline important? Is comfort important or you just need to get there?

If you just want to get there with no restrictions, typically airlines in China have cheap fares to USA. Once you reach USA, take a domestic flight on their budget airlines like Spirit and/or Frontier.

I cannot assure you of the quality of the travel experience but typically this is the cheapest way I can think of. But it means breaking up the booking so you have to manage quite a few things.

Another way is to check the United website, and every leg of your journey will be covered in 1 booking.

Hope this helps.

It depends on when you are leaving and coming back (holiday season? Off-peak?), as well as which day of the week, how soon you intend to fly, etc.

With some basic inputs, the cheapest round trip I found starts from ~$1200/pax. I'm sure there are better prices potentially available if you are flexible with your dates.