I haven't been to Korea so I can't compare it with Singapore. However, why are you coming here? I guess that's the big question and maybe people will be able to better answer your questions.
I'd suggest only coming here if you were offered a high-paying job and your moving expenses paid. Cost of living is very high compared to the USA, but I don't know how it is compared to South Korea. Check for online cost of living comparisons internationally.
http://forum.singaporeexpats.com/ftopic89846.html
Apartment costs will vary depending on where you live and what type of place you get and whether you're near a train station, near the water, etc. You usually deal with a private landlord here, as opposed to a
property leasing office. Some landlords are tough.
Also, if you do come here, I hope you have an undying love for the following:
Crowds
Extreme heat every day of the year
Long lines (long Qs)
Overpriced items
Trips to the grocery store that last a good chunk of the day