The chain link fence between my neighbor's property and mine needs repair. How can I tell who owns the fence?

Who owns the chain link fence that separates my neighbors property and mine?

Have a survey done. It will show boundary lines and then you can see where the fence is located.

One clue might be which way the fence 'faces.' If the pipes are facing you, then the neighbor installed it. That's tradition anyway, but not etched in stone.
It's also possible that both owners chipped in on it, and it's right on the line.
Best option would be to have a friendly chat with the neighbor about any known history and all possible options.

You can pay for a cheap survey, but it's one that you can do yourself. Go to town hall and get a map of your area. I paid $2 for one. Then get together with your neighbors on all sides with a lot of tape measures or laser measurers. Also look at your deed. It should have a map of exact feet and inches of each of the 4 sides.

A 'real' survey with transits and nearest USGS marker is expensive. That's the only way to place you exactly on the earth, with latitude and longitude.