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We have a polo and a dog. We like camping, have sups and bikes. We have a roof rack to carry DIY stuff it bolts in and is semi permanently fixed.
In this situation would you get a box or a bag the awkward shape of things like the paddle boards kinda makes me thing a basket/platform thing might even be useful.
We will need a new car but that's not on the cards until prices and lead in times stop being crazy.
Go for the biggest box you need. Maybe a long/thin one then you have space for other things like a couple of bike racks.
A bag would probably be too easy for the dog to escape from.
Have you had a look on roofbox.co.uk to see what the options are?
Is a trailer an option?
Thule Ranger ‘soft’ roof box will take a couple of inflatable SUPs - folds down smaller than a rigid box for storage
Think you can hire them to try before you buy?
I saw a pretty persuasive video for basket type racks, if you don’t mind packing in dry bags. I don’t enjoy the packing Tetris of a roof box, and the repacking when the lid won’t close.
Basket much easier to store off the car too.

Baskets are flexible for camping but it's a bit faffy. Expect some 'clambering'.
Roof bag here. Easy to store 🙂
Fischer. Excellent quality and rather less money than thule.
I got a roof box years ago and it's been well handy. My only regret was not thinking about its storage. Too big for attic, don't have a garage so it takes up half the shed for 50 weeks of the year!
Got a roof bag Thule ranger. Packs away neat and small BUT you can drive round with it on empty so has to come off when you get there.
And with that onehundredthidiot kills a bag dead in the water, hadn't considered the empty issue. **** that. I have a 4 room basement so storage isn't an issue!
Looks like I may have a lead on a box which should work nicely.
A cage would look baller though.
Not that I've tried a basket or bag, but happily using a BIG box on our van. Gumtree/FB marketplace throws up some bargains for roofboxes - got our Kamei one for £100, it's around 550 litres I think.
As you've got the storage, box all the way.
Thule Ranger bag takes a couple of minutes to install/remove - four locking clips and U-bolts, no tools needed
It's already 350 notes which is fast approaching the current state of the car!
Any level of removal is an additional pain in the hoop though.
You can get half baskets which would leave space for bikes or DIY stuff and probably about the perfect width for large dry bags or pelicases.
I have a garage with a pitched roof, so the roofbox lives on the rafters most of the year. It'd be a colossal pain if I didn't have somewhere convenient to store it...
A cage would look baller though.
Well yeah, in an Icelandic superjeep sort of fashion.
Uk/Polo = box.
I’ve got a long half width Thule roof box. Works really well for how we use it, but storage and mounting is a bit of a faff. It’s an older model so the clamp fixings are a bit primitive.
I also bought a “pick up truck” type roof cage with the intention of cutting it down width wise to fit some body boards and a SUP. Didn’t measure it properly and it’s about 50mm too short.
A long roof box might end up looking like a 747 piggybacking the Space Shuttle, rather than the other way round, especially on a Polo.