Gugu, if the landlord ask for the SIN, it's not to write it down and frame it on the wall: he will use it. It's illegal to use the SIN for anything that is not related to government business and leasing an apartment isn't government business. Saying "He can ask but you don't have to tell." is equivalent to your employer asking you if you are gay. Why? Because not answering will be seen as a red flag in both cases. Certain things don't have to be explicitly written in the law, they are implicit by association.
These days, the SIN is totally irrelevant for a credit check anyhow. All they need is the name and birth date. Previous addresses is a plus but not a must. Why? It's illegal to use the SIN for any business that is not related to the government. This mean the SIN should not show-up in any credit check report and should not be used by credit check companies.
Some companies are even doing credit check as a condition of employment and it's legal in Québec. (I checked since I faced this situation.)
Getting somebody's SIN is one of the first step to steal one's identity. Nobody should give his SIN to anybody who is not legally authorized. Imagine the landlord, with his collection of SIN and names from previous, current and to-be tenants, their employer name, past renting history, phone number and other stuff. This person might be the most honest on earth but, what if he get burglarized? All that good info could end-up in bad hands. Besides the legal issues, it's not only about giving the guy your SIN, it's about how well it will be protected.
A big insurance company (La Capitale) had to spend a huge amount of money many years ago to change their filing system because they were using the SIN as a file number. They were not even using it to get info about customers, they just figured, since it was a unique number and all Canadians have one, it was perfect to identify every customer, without any risk of confusion. If a corporation have to abide by the law, you bet a simple landlord must do the same. The SIN is for government use and communications, nothing else.