I heard that when a spur is put in it was the person requesting the spur to pay for it...so it should be private property, right?
Tricky, here.
If truly abandoned and on private property, a title search will be the determining factor. That having been said, I would think it then belongs to the property owner.
I guess what I'm saying is, you'll not get a definitive answer in this forum.
A good place to start will be the Tax Collector's office for whatever county or parrish in which the property is located.
But that may not be good news. There may be environmental issues at play. Road bed is usually full of bad stuff, at whatever time in service... creosote, oil, grease and other contaminants may have to be accounted for.
Edit: I neglected to mention the possibility of that railroad track being part of an easement for the railroad across private property. I have such an easment on one end of my property wherein the UP uses the western most part of the property as a season creek runoff. I've lived here since 1986 and have never seen a drop of water in the canal of the easement. At the other end of the acreage, my neighbor to my north has an easement for access to his property across my property.
But it is there, and a part of the property and records show it.
Might be better to fly below the radar here...