Who owns the propane tank if there is not written tank agreement?

Look for a decal on the tank! I went through this. No one told me and I had never had propane before and assumed that I owned it.

If a propane company owns the tank, that's all they need to prove if it comes down to a dispute. The burden is on the buyer to know what was included or not with the property. If you don't want that company, simply tell them to come get it when it's empty. You don't 'owe' them anything.

Is it a large tank, and is it attached to the property? Assuming it's not one of those little tanks that are used for BBQ grills, etc. (and I can't imagine anyone arguing about one of those) and further assuming that there is no label on it, it may come down to whether it's what's called a "fixture".

I just don't see one being claimed as a fixture any more than a garden hose would be. The only attachment spot is a threaded 1/2" gas line.

The odds that you own it are slim. It would take a pretty dumb seller to omit mentioning it. The tank, installation, and gauges are expensive.
You pay slightly more for propane when the company owns the tank, but it's worth it. You want to call around for help if the gas line breaks or the gauges break in the dead of winter?

I have no agreement but bought my own tank, 13 yrs. Ago. I use the same company and the difference in cost has paid the tank off. Who supplies the propane, they should know.

How do you know for sure there is no agreement? Did the seller state he owned the tank? Open the ball shaped lid and look for a tag.

The issue will be proof of purchase, if there is a dispute between the tank company and the home owner. The original owner of property should know if they own it or not. I bought mine, so I was free to order gas from any company I wanted and could work on lowest cost.

There had to be paper work somewhere, at some time that explained it. If no one knows, contact the tank company, and ask about repair on it, and see if they claim it ?