It is an interesting question and the best person to answer this question would be a public health expert.
In Singapore, only 2 vaccines are compulsory by law - diphtheria (part of the 5-in1 DTaP vaccine) and measles (part of MMR vaccine). The other vaccines that are part of the childhood immunisation schedule and are required for Primary 1 registration are: bacillus Calmette-Guérin (BCG), pertussis, tetanus, polio (part of DTaP vaccine), mumps and rubella (part of MMR) and hepatitis B.
Other than the vaccines above which are given free to Singaporean children in public health sector, all other vaccines are not mandatory. Compulsory vaccination for any disease is difficult. It is an ethical issue balancing public health and personal liberty (freedom to choose). The disease has to be significant enough (either causing deaths or widespread harm) to justify a compulsory vaccine. The cost of vaccinating everyone has to be considered too.
One can possibly argue for a compulsory vaccine against influenza or pneumococcal disease too. Along the same line, one can also argue for a blanket ban on smoking in the interest of public health and "herd immunity" against certain cancers linked to smoking.
With regards to chickenpox vaccine, the take-up rate in Singapore is probably not high. One of the reasons is that it is not part of the recommended vaccines to be given in the childhood immunisation schedule. In fact, up till recent developments (which I will elaborate upon), chickenpox vaccination for a child is not Medisave claimable. There are also many people who feel that chickenpox is not a serious disease and it is just a rite-of-passage that all will go through.
Good news is that all these will change come Nov 2020. Chickenpox vaccine has been included in the childhood immunisation schedule from November 2020 onwards. And the government subsidies for childhood vaccination (including chickenpox) will also extend to CHAS GP clinics. (https://www.straitstimes.com/singapore/enhanced-vaccination-subsidies-for-eligible-sporean-kids-adults) Previously Singaporean children can only get these vaccinations free at polyclinics. With the inclusion of chickenpox in the immunisation schedule, and the increased accessibility of vaccination in both private and public healthcare sectors, the level of chickenpox immunity in Singapore will increase.
