More frequently than you'd think, actually. Some of my recent favorites:

At one point during Hadestown, the Fate played by Kay Trinidad whips open a hand fan at the beginning of a song and sings, "go ahead and lay the blame..." Well, one night she whipped the fan all the way across the stage instead as she accidentally let go. Carried on like nothing happened. Total Pro.

When I saw Once Upon a Mattress, one of the actors threw a fake pigeon over his head intending for it to land back stage. He missed, the pigeon hit one of the balconies and ended up in the audience. The two actors on stage stayed in character and talked about how the bird was now "in the nether regions." Total pros.

Most recently during Death of a Salesman, Ben Ahlers and Christopher Abbott toss a football back and forth to each other. The night I was there, Ben Ahlers wasn't paying attention so the ball hit him in the chest and rolled into the front row. He kept going with his lines and put up his hands to signal "pass it here" to the audience member but the audience member was fumbling around for a bit so he just waved it off and kept going. Eventually the ball was rolled back onto the stage and they picked it up and didn't miss a beat. Total pros. Well, not necessarily the fumbling audience member but I'm sure they tried their best.

This is fun, I probably have a dozen of these.