I look into the education level of immigrants to Europe compared with immigrants to the United States. The data is from the IAB Brain Drain data set. The data shows the number of foreign-born individuals of age 25 or older, and the values are from 2010. The education levels are Low: less than high school, Medium: high school, High: more than high school.
Immigration broken down by region of origin and education level: 
The same plot excluding Mexico in order to see the rest more clearly:
And a plot of the degree of selectivity in Europe vs US. Selectivity is measured as the sum of immigrants with high education level minus the sum of immigrants with low education level.
From these graphs we see that for most regions US is more selective in the preference for more highly educated immigrants. However, the contribution from the single country of Mexico alone reverses that trend.
The gender distribition from different countries:
Interestingly, the same gender pattern occurs for both Europe and the US. More men than women immigrate from Africa, the Middle East and Mexico, whereas more women than men immigrate from the remaining regions.