Where is the best place in NYC to buy unique handbags that are on trend but the whole world is not carrying or has the best overall selection of bags for women who are really particular about their bags.
It depends what you're looking for. If you want certain designers (e.g. Prada), I think Barneys has a pretty good selection apart from a Prada store. Bergdorf is also good for Lambertson Truex, Nancy Gonzalez, some Fendi (but not as much as the Fendi store itself). Suarez is also good for Roberta DiCamerino (only place you can get them in NYC other than Barneys) and also for their own line of exotic skins and other designs. If you have a particular designer in mind, it's probably best to go specifically to that store for range of selection. In the area around 57th Street, you can go a few blocks in any direction and encounter pretty much everything (well not everything, but enough to keep you busy)...
I also recommend you look around the 57th street and madison~fifth area. There are a lot of flagship stores there (Chanel, Burberry, Bottega, etc.).
If it's too cold to go in and out of places, you can look in Bloomingdale's and Saks. And of course Bergdorf (but I have to admit, that place kind of intimidates me!).