Unless you put some sort of grass barrier such as a metal or plastic strip under the edge of the fence, the roots of your neighbour's grass will just keep pushing through under the fence. Even if you decide to put pretty gravel there, I would consider putting down this type of barrier. With one foot of space, you could consider growing a narrow vertical grass such as variegated Forster's grass e.g.'Avalanche' or 'Overdam'. They will take some shade and actually do grow very stiffly upright without support.

I started these Virginia Creepers 2 years ago from cuttings of my neighbours. They are covering the wire of the dog pen. I can't believe how fast they filled it in and only about a foot from the fence. They get south/east exposure. Maybe you could tack up some chicken wire or a pea net. This pic was taken June 17th. They are now covering the whole fence.

{{gwi:538161}}

I like the idea of Virginia Creeper! Thanks for everyone's suggestions.

Now to add my 2 cents worth. Whose fence is that, yours or your neighbours? When we lived in the city, our lot already had a fence, our neighbours. I never even thought to ask him when I planted my Virginia Creeper. He did not like a vine climbing on his fence! He even threatened to go to by-law if I didn't remove it at once. We put up our own trellis and then he couldn't say anything. All I'm saying is, if it is your neighbours fence, just ask if he/she is okay with it. Avoid any hassles. If you choose the VC, remember, it will become heavy after a few years and need good support. It also turns an awesome burgundy red in the fall when frost hits it and before that it gets blue berries that birds love. Marg

Marric,

Thank you! I didn't even think of who owns the fence. I honestly do not know, but I'll ask the neighbour and look at our survey. Our home and the fence are not new.

I have grown hops on a trellis in the past. It was quite spectacular, too, but it attracted insects that were annoying like aphids and wasps or hornets.

...Pam

Here's another 2 cents worth...
I agree with the barrier between your neighbour's lawn and your side (can't stress this enough; been there, done that!)

I also agree about putting rocks or pebbles in such a small area. So...

Seeing that it's such a small space, and that you want to keep the walkway clear, and it only gets afternoon sun, how about some flower boxes or pots hung up with long trailers like lysmacchia numularia or silver beacon or vinca for an added effect? And seeing the fence is white gives you sooo many color options!

You get no weeds on the ground, you get vegetation that does not block traffic AND you cover an otherwise bland wall!

How's that for a bargoon? :D