At this moment, I would say my favourite shopping haul would be sneakers. Cheapest price that I've encountered is ABC MART in Japan! If you're looking for shoes, here's a great place to check out.
For Taiwan, I believe most people know the typical tourist places etc. Ximending, Shilin Night Market, Raohe Night Market and so on. You'll have to walk around to price check which shop is cheaper, *cough* if I get too tired of walking. I'll usually get it when the price is cheaper than Singapore one. If you watched my snapchat in Taiwan, I have actually talk about the price difference.
Nike and Adidas is crazily cheap there, I got most of my sports bras at less than $40. If the shop is new, they will even have a opening promotion. SAY WHUT?!
Another of my favourite haul will be mask, there's a current mask that I LOVE SO MUCH at the moment. It's selling at $5.20 (?) for a piece in Singapore but I got them at Taiwan for $2.85 a piece, but instead of selling piece by piece like Singapore. They come in a box or a bundle of 3! Thank god!