Aww I really feel you on this, it can be quite overwhelming when people suddenly tell you "eh you need to pick a paed already!" when you're already juggling so many other things before baby arrives!
My neighbour brought her baby to Lion Health Clinic and said their doctors trained at NUH and KKH so she felt very assured, plus she mentioned all the standard vaccinations under the government schedule are fully subsidised for Singapore Citizens which was a nice bonus.
Some mums in our group love going by personal recommendation, a few others preferred to do a pre-birth consultation to "interview" the paed first before committing, both ways can work one, just see what you're comfortable with!