Not the NY I live in and enforced laws in. NYC enforces the state penal law ( 265.01) in all but the exposed clip and blade lenght ( which are NYC Admin Code). Any knife that can be flicked open and locks open ( even holding the blade instead of the handle) is a gravity knife in NY. Too much case law affirming this to list. Butterfly knives have been found not to be gravity knives in several cases due to the fact they do not lock open auotmatically, BUT they have been declared a dangerous knife due to the fact they are weapons by design, and our former Governor went after many out of state knife retailers for selling them in NY via internet and mail order and won lots of fines. You will be arrested under 265.01 for a dangerous knife if you carry a butterfly knife.
The NY law now explained as it exist ( Not being sarcastic but I would rather tell you what happens in court and at two in the morning during a street/traffic stop than make someone feel they are OK to carry a certain knife).The OP asked what is the search right of a officer. If you have a knife in plain sight ( a clip exposed, sheath, handles sticking out of your pocket, and an outline of a knife clearly visible - think that old 110 in your rear pocket of a pair of Levis that is clearly a knife), then an officer has the total right to question you as to the type of knife and how it functions, and yes take it from your pocket to see it. He does not have to offer in court that he had any reason why he stopped you, other than to say he clearly observed a possible weapon violation and acted to investigate it.(The rear pocket carry is the only one that may have some grey area. ) The good thing about NY law, is that you can carry a knife concealled ( the Sullivan Law has been off the books since the 1960s), and that does aid in your defense as to why you were stopped and searched.
I always offer the safest way to act as I have seen too many people have their lives ruined by a serious criminal charge due to the fact that they were ill advised about the law, or thought that since a store sells it you must be able to carry it. Please do not take anyone's posting telling you to "do what you want" as good advice. Understand the law, decide what your life would be like if you were the one in a hundred that was arrested for that charge, and make a mature decision that you can live with. As for me, I am looking to move to New Hampsire some day so I can legally carry any knife I want, but till then......