Try Madras New Woodlands ([Singapore] South Indian breakfast at Madras New Woodlands) on Upper Dickson Road, which is my personal favourite, or Saravanaa Bhavan (https://saravanabhavan.com.sg/) on Syed Alwi Road, both in the same Little India neighbourhood.
A branch of Bangalore’s Mavalli Tiffin Room (MTR) is also worth a visit.
Of course, one cannot talk about South Indian breakfasts in Singapore without mentioning the grand-daddy of them all - Komala Vilas (https://komalavilas.com.sg/public/home) - it has a whole Uthappam section to satisfy your every whim - tomato uthappam, onion uthappam, ghee uthappam, etc.
