Hallo Bobby and welcome to GOY. Sorry to hear about your dispute - always a difficult one.

Be careful about quick growing plants, as they have a habit of getting out of hand - and then don't stop where you want them to! You don't want them scrambling over the fence and making more upset.

As you've had baskets there, I'm assuming that there's enough sun on this area...so you could plant winter-flowering Clematis like C. cirrhosa, which is evergreen. There are a few to choose from. Plus maybe others for the summer, and a non-vigorous climbing rose, too? You'll need large containers, though.
If you wanted to carry on with your hanging baskets, why not use large posts to hang them on, in front of the trellis? That would draw your eye into your garden, too.
I'm sure there are lots of other climbers to choose from - but do check the ultimate spread and height.

If it's a panel fence the biggest problem would be the climbers creeping through the fence and your neighbour grabbing a handful, damaging them on your side too.
So how about Fuchsia Lady Boothby (which can be supported by canes), Hollyhocks, tall ornamental thistles etc?
A very rapid and unusual annual climber is Asarina, but it will find it's way though the fence panels!
Alternatively, let things cool down and you may encounter a fresh attitude next spring!!

Um, unless I've misunderstood, are you not saying that you will have a trough with a trellis attached to it for plants to grow up? If that is what you're saying, no climber will tolerate being in such a shallow container - contained climbers will need a pot that's 2 feet deep by 12 inches wide as a minimum. Why not set a fence post, or two, of your own on your side, and hang your baskets off those - or stretch trellis between them and grow something in the ground up the trellis, always assuming you have soil there.

Why not plant in the ground and place some tall Obelisks or Pergola posts with thick hanging ropes between, then perhaps things like clematis would be happier?

Roses like that, too - the RHS gardens have those, and they look great!

Yes they do Spritz the ropes are finished off nicely at the ends too.

I visited an open garden where they'd done that - it turned out to belong to an RHS committee member! LOL. I can't blame him for copying the idea, though. I would if I could!

I have some large pots in front of a fence (ours). they are on wheels. When they are planted they are 3 or 4 feet tall when in full bloom. I used to have hanging baskets on the fence, but I like to move the pots around the patio for flexibility if a planting is not happily placed or I fancy a change. I notice our neighbour has put some screws through our fence. As we are friendly it doesn't worry me. I do not think heavy pots dangling on fences, ones own or not, is a very good idea,as they can get so heavy. If this is near soil then Drc has a good solution. My Crinum is taller than the fence when in flower!!!!

Let's hope the fence doen't collapse, then, Dorjac! At least it'll fall their way....

Iv'e been round there watering, and its a bit of trellis for a planter, which is on top of a low wall which is their wall. I do know he discussed it with the people on the other side before they put up trellis all the way down. Still they can change their minds!!!! So many things for neighbours to squabble about.

welcome to Goy Bobbie. Instead of the trough a raised bed which is bottomless and will sit on the ground will allow roots to go through in to the soil below. You can have 6 or 8 ft posts with a trellis, although I do like the rope idea. I have seen that and it looks lovely. Sorry to hear about the neighbour problem. Can I ask how long is this fence you want to disguise.Our neighbour has put up a very ugly fence which dips at one end making it look even worse and then when he turned the corner he put in completely different fencing. Grr. We have bitten the bullet and put in a hedge of what will be free flowering privet. It won't disguise the fence overnight but it will eventually get there. I have also planted a couple of eucalyptus trees, which are much faster growing.

Lets be fair about this each hanging baskets is a huge weight and permanently dripping/damp putting quite a lot of stress on fencing especially as it gets older. Nothing is worth falling out with neighbours. Lots of things can look good with out involving another's property. I have several clematis along the edges of my garden but the plants are 1ft away from my neigbours fence, the other side is my fence and the trellis was fixed to the fence posts, but this year I have moved that away from the fence by 1ft as I find they look and flower better, which is I suspect because they get more sun and water.

That is a good point Drc726. people often plant too close to a fence thinking they will get support from it when in fact all they are doing is ensuring the poor thing is starved of its share of water. We recently spent a considerable sum having a mutual fence redone because the next door neighbour had let her ivy grow through the fence. We agreed with the new owner to share the cost on the understanding that the ivy would go. I recently found the stuff forcing its way through again and a quick word about it brought about a satisfactory result. Good neighbours are worth their weight in gold.

I think so too SG