For handbags in both Singapore and Hong Kong, I would look at local brands for something unique. There are lots of imported brands too, but high rents mean that those are often cheaper to purchase in N. America unless you were to look for the European versions/styles that are not sold here.
My fave souvenir from Singapore is Prima Taste meal kits that can be found in almost all supermarkets. These are entire sauce kits that include the grated coconut and coconut milk pouches in addition to the spice paste for a very authentic home-styled dish. I also love the Pandan Chiffon cake. The bakery that is most famous for this has a shop at the airport so there is no big rush to buy one in the city.
For Hong Kong, if you love candied ginger, I would get the ginger cookies from Kei Wah Bakery shop at the airport. I also tend to go for dragon beard candy. If you're in town, look for Marks & Spencer for British treats such as cookies and shrimp cocktail crisps. If you're petite, the Marks & Spencer in Hong Kong features long-wearing career basics in petite Asian sizing at very reasonable prices. For something upscale and embellished in the silk scarf department look for a local brand called Kenkei (sp?). For basic silks look at Yue Hwa department store or Chinese Arts and Crafts store.
In Thailand, have a look at the Jim Thompson store. My favourites are actually their giant cotton voile shawl-sarong-scarfs rather than their silks. For silks, I like looking on the Thai Design floor of major department stores for high quality items. Another good souvenir would be pewter tabletop and household items from various high-end local brands. These can be found in the home decor section of department stores such as Central and Emporium. Styles range from ultra modern keychains and scatter bowls to ornate centerpieces. If you're looking for something more utilitarian, stainless tiffins from ZebraHead brand are the best. For food to eat on the road, maybe with a drink or two, the local cashews and seasoned anchovies are so good. If you like spa items, and fancy soaps, check out Thann.
Vietnam has become insanely expensive over the past few years. The only things I still buy there are toothpick holders made from cassia bark (cinnamon) and bespoke clothing. If you can, have a traditional Vietnamese pantsuit made to measure. It consists of loose trousers with a long mandarin-collar tunic-dress that is slit up to the waist. It is very flattering while still being comfortable and practical to wear.