Kindermusik offers music and movement from newborn to 18 months under Village programme and progress to Our Time after finish the first stage. My friend is the owner of Kindermusik with Love studio and is highly recommended. http://www.kindermusikwithlovestudios.com.sg/programmes/
Another programme is Julia Gabriel, there is a playgroup programme from 6 months to 18 months. http://www.juliagabriel.com/our-programmes/adult-accompanied-programme/
If you believe in right brain training classes, you can join classes such as Shichida and Heguru. However, these right brain class are expensive, ranging between $60-80 per sessoin.
For young baby, it is recommended to join parent-accompanied group instead of independent class. It is a joy to accompany and watch baby learns in class.
If you are looking for cheaper option, you can organiser/join playdates for same group age baby. You can sign up meet-up groups in Singapore: http://www.meetup.com/cities/sg/singapore/