Here is a list of the most common surnames in Canada as a whole and in the country's five largest cities. According to this CBC report, University of Toronto Prof. Jack Chambers argues that
Over time, the number of surnames has actually not increased with the population, which has made many of them even more common. Says Chambers: "We've got just exactly the same resources now as we had a thousand years ago in Europe, and 3,000 years ago in China when surnaming began."
Well, not exactly. I don't think he's taken into account cultures where surnames are a quite new phenomenon. When I was growing up I had a lot of relatives whose surnames I didn't even know, and it never occurred to any of us in my immediate family to ask. Surnames were simply not all that important on that side of the family. Many had been taken on only with immigration to the US and elsewhere.
My father was born with a patronymic only. When he went to school the authorities wanted every student to have a first, middle and last name, so he took on his grandfather's surname, which may have had French or Turkish origins, and grecianized it to Κοϊζής, later anglicizing it to Koyzis. This means that my own surname is only about ten years older than I am.
In the Dutch-Canadian community with whom I work, I have long heard that Dutch surnames were chosen only about two centuries ago when Napoleon forced them to take them on. The Dutch often chose humorous or tongue-in-cheek names, assuming this was a passing fad and would never stick. Two-hundred years later their descendants are still saddled with them.
As for the Koyzis name, due to the paucity of married males to pass it on to future generations, it may end up dying out altogether. So perhaps Chambers is right after all.