Bagging seed heads - what bags?

I've tried many versions of netting over the years - built "proper" cages over the lot with poles and netting............but the birds would creep under the netting and get trapped, or catch their feet in it and have to be released.
Net curtains have much the same problem, birds go underneath them.
Also, I need to be able to get to the fruit to pick it, so anything that stops birds, stops me too.
The woodpigeons are jumping on the branches and snapping them - its not been a good year for small fruit - but a good year for cuttings.