it depend on how big/ old it is, where it is in relation to fences/building etc. a chainsaw seems most likely but do you intend to do it your self?
if it is a certain size it may have a tree preservation order on it which may make it illegal to cut it down without permission.

sorry to just land questions back at you.

TIf they are indeed "hundreds of feet tall" then they are mature trees - Sycamores grow to about 100ft and will take 30-40 years to do so. So you will not kill or even stunt them by spraying with any legitimately available herbicide. The only thing that is going to kill them is a tree surgeon with a chain saw.
As SBG says, they may well have preservation orders (or Ireland's equivalent) on them to stop you from chopping them down.
I'm not sure what you mean by the snow taking longer to melt because of the canopy. There is no canopy - all the leaves have fallen in autumn. I would add that you will probably find the snow lasts a lot longer and the wind is a lot stronger without the limited shelter that the trees provide. Removing 20 mature trees in close succession could actually cause you land heave problems as they will be accounting for a large quantity of water which, otherwise, will just raise the water table.
All in all, in a rural environment, I wouldn't advise removing them - and I live with an urban sycamore right on my front door step.

I'm intrigued as to why you think the electricity company has any responsibility for these trees which are on your land.
And the horses are their owners' responsibility - if the owners have chosen to put their horses in the field seeing sycamores about, they are probably aware that the risk is manageable (ie by moving the horses for a couple of weeks in autumn and again in spring)

They were there before you purchased your house and before you were born. I can't understand why this is bothering you now since the conditions which you now consider problems existed long before. The snow accumulates heavier around them because the tree and branches act as a wind break causing the snow to settle faster around them. It sounds like they are thriving and to take them down might cause more problems than you have now since everything is in a delicate balance. Girdling a tree or poisoning it is foolish because the tree is not going to dissolve away and you still will have to have them cut down and taken away which is a very very costly undertaking. Lastly, Urbanite mentioned the water table. Here's an example of how trees affect it. If I wade into one of my favorite fishing streams early spring I am in waist deep water. Once tree leaf out occurs late spring that waterlevel in the stream goes from waist deep to below the knees in less than a week, thats how much water these trees draw out.

Couple things:

If the trees are across the property line, you should secure permission and agreement before removal. Property owner should pay some of the cost to remove the trees.

Don't even think of taking down the trees yourself. You will kill yourself or somebody and/or do major property damage. What will you do with them once they are fallen?

Call in a professional arborist/lumberjack to do the job. Maybe you can sell the lumber. Girdling the 100+ foot tall trees or using herbicide will only weaken them - causing them fall or drop limbs in high winds - too dangerous

what does the farmer say? as if I read your comment correctly you own the lane and trees jointly.

The only way to kill a sycamore is to cut it down and apply a stump killer to the stump. If the trees are as large as you describe then it is a job for the professionals. Sycamore is a non-native species and it sounds as if yours have got out of hand - I hope that you have a woodburning stove.

In the valley below us along sidethe brook is where the electricity poles are situated, one year we had so many power cuts due to the branches hitting the wires the company had to send a gang of men with a 'cherry picker 'to sort them out
now we get a low flying helicopter in the autumn checking all the overhead lines in the area

maybe you need an assesment by professionals beforedoing anything drastic

If the trees are not rooted in your garden you have no legal right to do anything more than remove any branches that overhang hour property. Our garden also has a lot of mature sycamores to the east and the west but we accepted this when we bought the house. The Electricity company will only remove branches that interfere with the power lines and this is not your problem in any case, and as has been said the horses are the owner's responsibility.

Try to think positive. The trees are a wonderful shape in winter. The ones on your land can be cut down and root treated as has been said. This is not a job you can do for yourself as has also been said above and could cost four figures for each tree. If you were to ring bark the trees that would kill them but then you would have a row of dead trees which would slowly rot become unsafe.

Best to research what will grow under them now you are stuck with them. Sorry to be so blunt but its better to be honest.