Is it common for service members to file their DD 214 with the local county clerk's office when they retire or leave the service?

Please do not file your DD214 with your local county clerk. This is strongly recommended against in your SFL-TAP process. Filing with the county clerk makes your DD214 public record and anybody can gain access to it, with positive or negative intentions.

If you're looking for a way to get your DD214 easily accessible, scan it to yourself and safe it somewhere securely.

CPT Jacob Swartout
In this day & age of electronic media, I personally don't see a need. I always keep a few paper copies on hand (stored securely).
I also have mine saved on external hard drives, a few laptops, desktops, & even the cloud. I also recommend keeping a copy on a thumb drive (but only if you don't have a habit of losing something like that). Thumb drives, while easily accessible and transportable, are also pretty unsecure (secure flash drives can cost big $$).

Absolutely ! It makes it a public record but more importantly it will be safe keeping in case you need a certified true copy or your family ever wants to research your career after your gone or in the event your incompasitated.

With the adoption of the DoD ID Number taking the place of the Social Security number, the amount of pertinent info shared publically lessens for newer retirees (when they decide to update the 214 to use that new number). However, if you rely upon the idea that you can always request a copy later, remember that there was a fire a while back in St. Louis that makes retrieval of my grandfather's records impossible.
There also comes the fact that you may have to file your 214 in the states that exempt military retirement pay from state taxes.
If you are concerned about privacy, check your annual credit reports and audit your credit card and bank statements frequently. Recovery from identity theft is a lot better today than 15 years ago. Most banking and credit agencies have an active fraud detection department (USAA comes to mind and has contacted me on several occasions) and policy to get you back on track quickly.

So I am not seeing how locking a DD214 in a safe protects the Veteran from public release of most of the information on the DD214.

https://www.archives.gov/st-louis/military-personnel/foia-info.html

I never did and see no need to. I keep copies of mine in a safe in a specially marked folder where I have consolidated key documents my family would need quickly if I pass suddenly. Plus electronic copies on my computer and a hard drive backup. I mean think about it--I would never put my will, life insurance info, etc on a public record system, so what makes the DD214 so special? There is nothing special or especially secure these days about a county courthouse records. The real issue here is how have you prepared your family for an emergency if something happens to you. If they have no clue where to find your key documents quickly, THAT'S the real problem, not whether you have it at home, or a bank safe deposit box, or a courthouse. Same with something like a medical power of attorney. You wouldnt put that in a courthouse for safekeeping. You'd keep that somewhere that was readily accessible by your family and make sure key people know where that is. Just put your 214 next to it.

I did. In case I lost my copy. That was in 1975.

There was a time this was recommended but in the digital age it is not something that is needed or recommended.

I have no idea why anyone would do this. It serves no purpose. With a little bit of effort, one can make sure a copy is always safe. If worse comes to worse, you can always request a new copy.

If you have Army Air Force Mutual Aide Association (AAFMAA) life insurance, they will keep a certified copy on file for you and/or your dependents. They will keep certified copies of other important documents too.