What are the country options apart from Singapore for a Master's degree with stay-back work permit opportunities?

Yes, like Singapore, many countries offer clear post-study work options. For example, Australia offers around 3 years for many Master’s graduates, the UK offers 2 years (currently), Canada offers up to 3 years in many cases, New Zealand offers 1–3 years, Ireland offers up to 2 years after a Master’s, USA offers 1 year of work permit (OPT) which can be extended for an additional 2 years for STEM graduates.