Any advice on how to establish a network and remain competitive in the job market for returning Singaporeans (2 years after graduating uni)?

Hope I'm not wrong by guessing you're in a finance related field since your internship was with audit. Nonetheless, what you can do is to reach out to headhunters that are specialized in the field to assist you on your job search.

overseas experience within the relevant field is always appreciated as you might bring a different perspective comparing to your other competitors interviewing for the same job opportunity.

update your CV and LinkedIn profile and start actively looking for jobs at portals that you might be keen in for a start! :)

It really depends on which industry and job scope that you really want to pursue.

Nevertheless,network is still essential part of career. There is no right or wrong way to start a network and it really depends on your job scope and industry: what type of professionals you will need to collaborate most in your jobscope. The most direct answer to start building network: go out and meet people related to the industry that you are pursuing your career in. You can start connecting with people: LinkedIn, seminars and webinars, hangout time with colleagues and coworkers

The key to stay competitive in the market is upgrade + innovation. If you are a Singaporean, you should have skills future credit , pick up courses that is related to your career and consistently learn. What employers are looking today employees is in value delivered; how much they can contribute to drive and grow the company, or deliver excellence business operations that drive business smoothly.