So we are purchasing a house with a decent sized yard. Just found out there are utility easements and they come out 14' from the fence on the east and south side. Some of the other neighbors have trees and sheds on the same easement. My plan originally was to line the fence with fig/plum trees. I am moderately concerned that it will be an issue since I have never dealt with this. What do y'all think the odds are that they will come dig them up? What would you do?
Here is a pic of it. The easement is in red.

I worked for a utility company briefly. The own the utility pole and have no problem hacking down anything in their path to get to it. Just a word of warning. Since you have above ground lines and the cost to bury power is huge I doubt they will dig. I would just make sure they have a clear path to the utility pole.

Thanks Livetaswim!! I'm thinking I'm going forward with my plans of lining the fence. Worst case scenario I'll have a bunch of cuttings to offer.

Why ask us when you can ask your actual utility company?  :)

Utility company will just answer boilerplate response. They own that area so they can do whatever they want. I think the best suggestion is to put your less loved figs close to the pole and the loved ones further away. Though perhaps there is something special about a utility pole that helps stuff grow well. When I worked for them I had a company issued machete to hack away blackberries which tended to grow best around a pole. 

You might also use the "call before you dig" service, usually 811, to have them check to see if there are any buried utility lines in the at easement.

I would warn against eating anything that may have absorbed something from the pole. They are treated with some nasty stuff. Every pole has a brand about 5' off the ground. First line is the manufacturer, second is the year and third is the treatment. If it has a C its treated with creosote. If its a P its pentachlorophenol and if it says CCA, thats copper chromated arcenate(the pole is usually a pale green). There are others but those are the three main preservatives used in pole treatment. All of them are carcinogens and they leach into the soil. I used to work for a pole retreatment company. We usually started retreating poles after 20-30 years of service.

I have a 5' easement for power line to my house. They will occasionally prune if the trees get within about 5' of power line. The only time there would be an issue, likely, would be if the pole needs replacing. They have even replaced my power line drop once without bothering my trees.

If you have power lines that service more than just you house, there might be more issues. It depends on how difficult it would be to replaces the poles/lines that run through the easement and what the likelihood of that needing to happen is. If there are no actual wires or poles across your property, it may be an abandoned easement or one that was never used.

I was part of a project to inspect all the poles in the city of Seattle. Most were relatively new, but some had been there since the 30s, this was just 6 years ago. I wouldn't worry about them replacing a pole unless it was visibly falling over.

You should check with the utility. I did not notice any actual poles in the easement area. If your property is in the back edge of a subdivision it may be that the easement was put in for future expansion of the subdivision. If there is no plan to expand the subdivision you may be able to buy the easement back from the utility and add it to your place. I was in that situation when I was in Florida and my neighbor had just bought his easement for a very reasonable amount from the utility. Then you will not have to worry about using it if you buy the easement.