Expats are usually either on a work visa (WP, S-Pass, EP, LoC), which allows work for only one sponsoring employer, or on LTVP/DP(no LoC), which allow no work at all.
In both cases, they cannot legally do additional informal or part-time work as you are looking for. Only PR and citizens can.
Asking on behalf of friend: Is it possible to expat husband and local wife to apply for an International school for their child?
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For reference the approval has to come from MOE and they refer to international schools as the Foreign School System (FSS). That's the term you need...