Another recommendation for Bata shoes. It's cheaper than your USD25-30 price range, but the quality is decent, especially the Bubblegummers range. There is a huge Bata at Peninsula Plaza (Opposite St Andrew's Cathedral).
http://www.bata.com.sg/pages/Find-Yo...d-1082084.aspx
Orchard Paragon Level 5 have many children's shops, but the price point is definitely higher than USD30 that you're looking at. I've posted some Singapore prices for you for your reference below and these are the types of shops you'll find in Paragon.
https://www.striderite.com.sg/
http://www.pediped.com.sg/sale
Another place I would check out is IMM. It's an outlet mall in the west side of Singapore, but there are outlet shops, good for getting shoes/clothes etc for cheap. There are several children's clothing shops in Level 2 (Pumpkin Patch, Chateau de Sable, Cotton On etc), and also adult clothing shops in Level 1 (Espirit, G2000, Coach, Sacoor Brothers, Timberland etc). Good for shopping for the entire family in air conditioned comfort.
(Personally though, I buy my todd's shoes online on Carters/Osh Kosh and have them shipped to Singapore).