As the guy who has been caught by each of the big three, I will provide you with some anecdotal information. All of these instances were while shipping on my online print-at-home account, for reference.
USPS - I will always ship with USPS. When I was caught, we had a phone call about the situation, and they sent me a form saying how it's federally illegal to ship alcohol blah blah blah. That "fill in the blank" form was filled out with a pencil. They ship coast to coast the fastest while only being slightly more expensive than UPS.
UPS - They're slow but UPS Ground is cheaper, usually, than USPS. When they caught me they just told me not to do it again, and I also got a letter in the mail. There letter was more formal that the Federal government's. I have shipped with them again, after they caught me, without issue. My account is in good standing.
FEDEX - They're on par with UPS in regards to pricing, but if/when you get caught, kiss your account goodbye. They'll also flag your ass in the system, so any time that you try to ship anything that makes a sound, they'll tell you that they're opening the box to inspect before shipping. I don't waste my time there any more.