There are both budget and non-budget airlines that fly direct from Singapore to Taipei.

1. Singapore Airlines 

The 3rd best airlines in the world (based on the rating given in 2014) have daily flights to Taipei. Normal price during peak period for this flight can reach S$1,000. Do keep a lookout for the promotion, especially the one-for-one Mastercard promotion for a direct flight to Taipei.

2. EVA Air

The Taiwanese carrier also serves daily flight to Taipei. However, timing-wise, the return flight from Taipei to Singapore is quite early in the morning, around 7-ish. On a brighter side, EVA Air offers a flight on Hello Kitty jet from Singapore to Taiwan. Kitty fans rejoice!

3. China Airlines

Republic of China's carrier which is also the largest airline operator based in Taiwan also has daily flight to Taipei. Their timing is better than EVA Air.

Budget Airlines (Direct Flight)

1. Scoot 

A subsidiary of Singapore Airlines, Scoot also serves daily flight to Taipei. If you prefer to go to Kaohsiung directly, Scoot offers direct flight on alternate day. My friend managed to book a return flight at only S$200+ during one of Scoot's promotion.

2. Jetstar Airways

Another budget airlines that fly directly to Taipei is Jetstar Airways, a subsidiary of Qantas Airways, an Australian-based airline operator. During non-peak season, a return trip to Taipei from Singapore can costs around S$300+. High-season like the long National-day weekend can costs you S$500+ though.

3. Tiger Airways

If you are not into Scoot and Jetstar, you can also opt to fly by Tiger Airways. Just like the above two budget airlines carriers, Tiger Airways also have two timings everyday for a direct flight to Taipei. The ticket price is around the same.