Late to the conversation, but shooting grackles/starlings is what we do for fun, entertainment and population control.
Shotguns are great, for practice and points games. But if you have a 'dark' sense of humor you can try an air rifle. Certainly less than what most people would admit to, but it can be really fun. The first option is to hit them just at the bottom of the throat. If done correctly, you will hit the spine with enough force to break it while leaving the bird in one piece. If hit right, the toes will tighten and the bird will hang upside down for a few seconds to a minute! For those skeptics, I have done it a few times.
The other option is to load the rifle with the pellet in backwards. Wad cutters work best for this. When it flys out the skirt will want to open up and upon impact it does. For best results, use a high power air rifle and aim for the center of the chest. Once again, if done correctly, the bird will be de-feathered with the body flying out behind a cloud of feathers! Usually we would use this for sparrows, but with a high power air rifle it works on the bigger birds.
I used to do both of the above as it was a part of my summer job.
Aside from that, the standard 22LR and 17HMR can make a good choice.