Singapore is a single jurisdiction, so the variation that matters here is not by canton or state but by destination country and airline, and a law-firm-grade letter is drafted with that in mind. For travel into the Schengen area, several member states expect a signed, sometimes notarised, consent letter from the non-travelling parent for any child crossing with only one parent, and they are far stricter than Singapore is on departure. A letter that satisfies a casual ASEAN land crossing may be waved through, while the same trip into France or Germany draws a request for notarisation and certified translation.
Malaysia and the regional land and sea crossings deserve their own thought. The Woodlands and Tuas checkpoints see enormous family traffic, and while a routine one-parent day trip rarely needs paperwork, a longer stay or travel by a non-parent is smoother with a letter in hand. Air travel through major hubs adds the airline layer: carriers running unaccompanied-minor programmes impose their own forms and handover rules that operate independently of immigration law, so the consent letter and the airline form do separate jobs and you usually need both.
For destinations that are Hague Convention contracting states, the consent letter carries extra weight because it is the documented proof that removal was not wrongful, which is precisely the question a return application would turn on. Where the trip follows a Singapore divorce, the destination's officers may also want to see the custody position, so the letter should align with the court order rather than contradict it. Parents finalising those arrangements often pair the travel letter with our Singapore divorce settlement covering HDB, CPF and maintenance so that the travel terms and the custody terms tell one consistent story.