Dog breed popularity has been primarily determined by fashion rather than function.

As for the relationship between breed health and popularity:

"We find a strong positive correlation between a breed's popularity in the U.S. and the number of inherited disorders from which the breed suffers. .... We also find that breeds with more disorders have decreased in popularity in 19962005, a result that parallels a similar, albeit weaker trend observed in U.K. data. Altogether, these findings suggest that popular breeds carry a significant health burden, and that in recent years the public may have started to avoid breeds with more health problems. The latter effect is statistically significant, but not large: number of inherited disorders explains only about 10% of the variance in popularity changes in both the U.S. and the U.K. (as estimated from ANOVAs with popularity change as the dependent variable and number of disorders as the independent variable). Indeed, number of disorders was still an excellent predictor of popularity in the U.S. in 2005, although it did not strongly correlate with popularity in the U.K. in 2007."