There are many shops in that sell second hand clothes in Singapore. Some have been featured on Recyclopedia.
They do not all accept donations. Some source their clothes from 2nd hand wholesalers.
Generally speaking, stores that do accept donations are EXTREMELY selective: there is far more supply than demand so they can afford to be picky. Be prepared for them to reject a lot of items.
But don't worry, if they don't take your stuff, there is always the Cloop Lifeline bins.
Shops mapped below include: