The "lead-crime hypothesis" is a very obvious example of correlation not implying causation.
For example, rock music causes crime. Rock music grew in popularity in the late 50's and early 60's, peaked between the late 60's and early 90's, and declined thereafter. Just like crime! In fact, the harder they rocked the worse the crime rate got. I guess Tipper Gore was right all along. Rock music causes crime.
Margarine causes crime. Johnny Carson causes crime. The Boston Celtics cause crime. The Japanese economy, big hair, VHF television, polyester, yearly Oldsmobile sales: all cause crime.
During the period leaded gasoline was sold, Canada allowed 0.77 grams of lead per liter of gasoline, nearly three times the US limit of 0.29 g/l.[2] Yet the crime rate in Canada was (and continues to be) far lower than in the US.[3]
In Mexico, leaded gas remained commonplace much later than it did in the US, and yet crime fell dramatically during the period leaded gas was sold, and increased dramatically after it stopped being sold.[4]
One thing we could look at is people who had the highest lead exposure. Lead miners, paint factory workers, house painters, fuel refinery workers, gas station attendants, auto mechanics, etc. I would confidently bet the farm that, adjusting for sex and income, there was no statistically-significant correlation between these professions and crime.
Aviation gasoline, as used in small propeller planes, still contains lead. We could also look at present-day flight instructors, airplane mechanics and people who live near general aviation airports. I would likewise expect there to be no relationship.
Here's something that's known to cause crime: young men. The Baby Boomers were the largest generation ever, which means between the late 60's (oldest Boomers entering their late teens) and early 90's (youngest Boomers entering their 30's), there was a particularly large population of young men. This doesn't explain the entire rise and decline in crime (the Boomers weren't that much larger than other generations), but it's a better starting place than lead paint.
[1] https://news.ycombinator.com/item?id=16025762
[2] http://www.columbia.edu/itc/sipa/envp/louchouarn/courses/env...
[3] https://ourworldindata.org/wp-content/uploads/2013/03/homici...
[4] http://www.slate.com/content/dam/slate/articles/news_and_pol...