Uh, not really. They still have drugs in Singapore but it has been pushed just a little further underground. There are plenty in the Singapore government who buy in that market, and so a little trade is allowed if it is kept low profile and you have some connections. It is the way these things have always worked in countries that severely restrict access to a desired product.
There are interesting studies in countries (e.g. the UK a century or two ago) where severe penalties are exacted for even relatively minor crimes. What tends to happen is not that the level of the crime is reduced but that people stop reporting crime and exact their own informal justice because they believe the official justice system to be inordinately or unacceptably harsh. This effectively disempowers the government to determine justice. If society does not believe the official justice system is fair, they won't use it.
What is the extent of the "drug problem" in Singapore and what is your source for this information?