Honestly, don't see much. One time, I was heading to my office (sometimes, I take the 57th bus to 5th ave and walk down) through Rockefeller Center, when I saw two ladies run out of the GAP on 6th (which is now closed) and point to a man with a rucksack saying he had stolen sweaters and the woman walking about 3 steps ahead and him was with him. A guy in a 3 piece suit and what looked like a highschooler, grabbed the guy's bag, and opened it, and sure enough, there were about 5-6 sweaters with tags on them. NYPD were standing about 10 feet away and pretended like they saw nothing. About 2 more random people came in and kept the guy from his bag until they got all the stolen merch from his bag. Once the ladies had everything, the guy in the suit handed the bag back to him, and everyone walked away. Guy and his lady accomplice scurried away, and everyone went back to thier business.
That is really the only things I have ever encountered in this city, other than random drunk people falling down outside bars (and having to help get them in an ambulance).