Isn't it interesting, then, that when you look at the list of common American surnames, which are often associated with occupations, that "Weaver" appears at position 190 (.05% of the population) but "Spinner" appears at 1/50th the rate, at position 7393 (.001%). Carder is at 4255 (.003%); Carter is, I would assume, a different profession. [The first 10 names, btw are: Smith, Johnson, Williams, Jones, Brown, Davis, Miller, Wilson, Moore, Taylor].
I'm not attempting to claim that there's some direct relationship between last name frequency and historical occupation rates, but it's still entertaining to speculate on the cause. My initial suggestion was that carding and spinning were more likely to be women's work and of course in the West women's surnames don't get propagated. Mrs. Guesswork suggests that spinning and carding weren't professionalized the way that weaving was [prior to the invention of the spinning wheel, spinning technology was extremely low-tech], so you might spin or card in your spare time, but weaving requires enough capital equipment that you would expect it to be done professionally and thus be more likely to get a surname attached to it.
Equally likely, of course, is that it's just coincidence, but what fun would that be?