As far as the major (most popular) handgun calibers go, how easy is it to usually find each one
Depending on how close you are to certain political elections, easy to impossible on ANY caliber, .22LR all the way to .45ACP
but want to know that my selected chambering is common and always available.
Buy based on some other criteria. You can always stock up whatever caliber you end up with and not come close to worrying about what the local Chino-Mart has in stock. Buy ammo online, it's cheaper and you can more often find what you want. Two years ago it was nearly impossible to find ANY caliber on store shelves - but those of us with a good stash already on hand kept the lanes at the local range busy regardless.
I hear that many or most police departments are now using .40 S&W, so that's a point in its favor
, but does that necessarily mean that I can always find it at Wal-Mart, Dick's Sporting Goods, or Gander Mountain?
By the way, I don't consider .22 LR a handgun cartridge.
That's good for me, and unfortunate for you.