Interesting point...
How many guys have you worked for in aviation who are of good moral character, yet they all hold ATPs!
No, certain convictions may well see you barred from holding a licence. I think getting caught selling, smuggling or using drugs would head the list but if you have lots of speeding tickets, for instance, you might well come across as someone who just doesn't care about the Man and his rules.
Anyway, here we're just looking at what you need to do the written (nothing much) and then to do the ride (whatever the book says you need). Don't cut corners and you will have nothing to worry about.