First off, thanks for noting the LegalBlade App availability. In asnwer to the OP's questions:

Note that New York City administrative code has an under-4-inch length limit and requires knives be carried concealed. Knife Rights recommends that you never carry your knife clipped to your pocket in New York City. Even when covered by a jacket, simply moving the jacket aside to get to a wallet has been enough to get folks arrested. Always ensure your knife is completely concealed at all times, including not "printing" on the outside of your clothing. In addition, be extremely circumspect about using a knife for any purpose in a public setting.

Note also that NYC has interpreted the state law against gravity knives such that if an officer can "wrist flick" the knife blade open and the knife blade locks open, that knife is an illegal gravity knife. NYC takes this position even if it requires multiple tries and use of exaggerated arm thrust or motion. Using this interpretation, most any lockblade knife might be deemed an illegal gravity knife.

A Swiss Army Knife, traditional slipjoint folder (Case, etc.) or something like a Spyderco Slipit if you want one-hand opening, all without a locking blade, cannot be considered an illegal gravity knife. And, of course, need to be carried completely concealed. If there's a pocket clip, take it off lest you by habit use it.

And, just to be clear, just because you cannot "wrist flick" open a lockblade folding knife does not mean a cop cannot. Trust me, they can and will. Since this is an arrest and go to jail offense, don't be stupid.

Added. BTW, just to make the point, I opened a Leatherman OHO blade in federal court: http://www.kniferights.org/index.php?option=com_content&task=view&id=373&Itemid=1