Every little Isabella I know is Bella at least part of the time, the same way every little Sophia I know is called Sophie. I sometimes hear Iz/Is, but I never hear Izzy. It's mostly a combination of Bella and Isabella in full. IMO, for this new generation of Isabellas, Bella has got to be the most common nickname.
I hear Bella a lot, but I can't always tell how many are just Bella and how many are Isabellas (or Arabellas, or Annabellas, etc). I would describe it as common and very "right now." Whether or not that makes it unusable is a personal judgment.
I do think it's a pretty name. It's actually my cat's name; my son named her after a canine Bella we met while taking a walk, and we meet other people with dogs or cats named Bella all the time. I think it's as popular for pets as it is for children, but I don't think that's necessarily a strike against it. Of course, I'm one who loves Lucy, Toby, Max, Daisy, and a bunch of other "animal names"...