How common are counterfeit alcohol beverages?

Actually pretty rare.   The only market for fakes would be in the high end booze area such as single malt scotch from Scotland.  And anyone who knows single malt scotch,  they would be able to tell a fake.

Try a bottle of Oban then try some Cutty Sark.   You will spot the difference right away. 

Virtually all top-shelf liquors have distinctive bottles and labels. If you are familiar with these designs, it is pretty easy to spot the fakes, because to make identical bottles and labels would cost the counterfeiter almost as much as the profit to be made from selling an imitation. As others have pointed out, it's more likely to occur in a bar, substituting lower grade booze for top-shelf brands.

I don't know that I've ever seen counterfeit alcoholic beverages.

Bootleg doesn't mean counterfeit, it means illegally made.   You have to have a permit to make alcohol - other than home brewing, etc.  If you want to sell the alcohol you have made there are regulations regarding commercial manufacturing and sales.  These regulations are for the protection of the public as your health can be harmed by some poorly made alcohols.I've had homemade moonshine (kicks your butt) as well as homemade wines and beers (mixed bag, I'll mostly pass).  

If you mean how likely are you to buy a bottle at a US store and it be fake, I'd say the odds are about zero that will happen.  If you are getting bootleg alcohol it is not likely to be in bottles that claim it is something else.

Depends on location. Use caution when price seems too good to be true, especially for

expens(c)ive, in-demand brands. I used to live  < 10 miles from Mexico so bought bottles 

in duty free shops regularly below retail, including duty. I was not however naive enough to somehow believe authentic Johnny Walker Blue should be under 100 dollars for a 750 ML 

yet typically going for about 200 in the US. Bottom line get familiar with product and prices.

ps: Beware of bars serving rotgut / bottom shelf product - especially where you might not NOTICE right away - from a bottle that originally had a higher quality one. One way to know they do this: Shots from a supposedly high-end brand at a much lower price than regular.

ps2: I love many things about Mexico and have family there. But you still have to be careful.

In the US it's pretty uncommon to get "counterfeit" brand name booze by the bottle. I think it is quite common to get inferior booze in a bar when you pay for a brand name and get something else that was decanted into the brand name bottle.

None if you buy from the store

I remember taking a cruise years ago, to Cozumel,, Mexico. At the dock was a Huge store advertising Kahula for $3 a bottle. The Cruise Director told passengers NOT to buy there. 

But of course some rich couple bought 3 cases. When they inspected it after leaving port, they found out they has bought 36 bottles of Mexican Tap Water-even that they couldn't drink. The glass the bottles were made from were worth more than what was In the bottles.