February seems to be one of the best months to visit Asia, same like November. It is a good period for the dry season and a good period for great deals.
Here is a flight deal from Istanbul. The bests about are the company, Quatar, and the transit times that are low.
Singapore is an expensive city, very beautiful but still impossible to visit if on a low budget. You can stay a night there and then explore Asia, because their airport is one of the biggest hubs for all Asian airlines.
I have done that one time for Bali. I also forgot to check my bag again as I have changed the low cost airline so I ended in Bangkok at return with no bags and clothes.
Enjoy and go search for low cost flights throughout Asia.