Wisteria, Honeysuckle, Clematis, There are many other choices. Depends on what you like.

Wisteria would probably be too vigorous and need constant pruning for such a small trellis. Honeysuckle possibly, but that, too, could find it's way in through any cracks, as could clematis. The latter is your best bet though. If you choose one of the summer flowering types then it would only need pruning once a year.

You don't say which aspect the trellis is facing. North, south?

What is the aspect? How much sun, if any, does it get? What soil do you have?

Why do you need to paint the trellis, though? If it's made from pressure-treated timber, it doesn't need painting or staining.

I have a clematis (allegedly evergreen) I bought it in Feb full of foliage and is brilliant, it gets mostly sun but also some shade and has an abundance of white flowers! It's even flowered once here already and it's only about a foot high as it's still brand new. Clematis terniflora robusta. Clematis Montana is a good one too (if a little rampant)

Both those are very vigorous though - the terniflora I believe can get up to 60 feet.... there are lots of smaller clematis though and it usually says on the label how tall they are likely to get. You need to consider the quality of the trellis as well, as it can vary a lot - a lightweight one won't support a heavy climber for long - eg a mature honeysuckle can get very big and heavy and pull a weak one off the wall after a few years. One of the smaller clems would be fine though, but it would not stay green in winter.
Another way to go would be to plant a pyracantha in front of it and just use the trellis to help train it flat to the shed wall - they can look very effective done like that and are semi evergreen and are happy to have bits you don't want trimmed off.And they have red berries in autumn..

However looking at your photo it seems that the back of the shed is largely out of sight, and a plant growing over the trellis could take up nearly as much time overall as painting the trellis (though Rosie has a good point - why paint it)

If you want the plant for its own sake most of the suggestions are useful, but if you aren't bothered about growing except for covering the trellis I would leave it alone.

Hydrangea petiolaris?

I have a beautiful honeysuckle called "Major Wheeler" with bright red trumpets. Hummingbirds are about it all summer. Pyracantha is aptly named "FIRE THORN" with good reason - they will even pierce leather gloves.

How about Cotoneaster horizontalis. It is one of the top ten recommended by the RHS. Its distinctive branches with their herring-bone structure are lined with tiny white flowers in late spring, and bright red berries in autumn. The tiny rounded leaves also turn red in autumn and, after they’ve dropped, the characteristic branch structure is more clearly revealed. Resilient, dependable and happy against a north wall. No pruning generally required, but needs tying in to climb well. 1.8m (6ft). I have it on a south facing wall in a howling gale quite often and it stays evergreen. The birds like the berries and the bees love the flowers.

Many many thanks for all your advice and suggestions.

My clematis has suffered from 'wilt' for the third time on my trellis and looked a very very sorry sight on Friday morning. I have absolutely no idea what is causing the problem but a few days ago it was healthy and robust and suddenly it looks totally dead. I have given up on the plant as I really want something that will give my trellis colour and grow steadily. We have been to the garden centre this morning and we were recommended Japanese Honeysuckle (Lonicera japonica 'Halliana'). My wife and I have just untangled one which was very reasonably priced at £12.99 and planted it. We also found some green velcro tape for attaching the plant to the trellis and it's brilliant stuff which can be easily removed and re-positioned rather than the wire or stapled tape versions. So fingers crossed we will have our trellis full of life in a few weeks time.