Fast Growing Trees
There are a great many more plants that will grow on clay soil, but these are consistent performers.
Flowers and ornamentals
Roses, grow really well in clay soils, they have strong roots that get through it, the clay then provides plenty of nutrients a good anchor against wind rock and a ready supply of water that the rose roots can go and find. Rosa rugosa - zones: 3-8S/3-9W
Phormiums, big trendy grass-like plants with broad sword-like leaves from New-Zealand. Similar reasons to roses, above. - zones: 8-10S/8-10W
Amelanchier, shadbush, white spring flowers and bronze new foliage
Aucuba, Cuban laurel, yellow spotted green leaves.
Berberis darwinii, thunbergii. Versatile and varied group of shrubs.
Chaenomeles - Ornamental or Japanese quince. Masses of beautiful red to pale pink flowers depending on variety borne on bare stems when much of the rest of the garden is still dormant.
Choisya ternata, Mexican orange blossom, Glossy evergreen shrubs that comes in two main types grown either for wonderfully fragrant white flowers in summer (and often again in autumn) with green leaves, or for vibrant yellow leaves in the variety "Sundance" which unfortunately rarely flowers.
Cornus varieties - Dogwoods One of the best shrubs for waterlogged areas for most gardens. Dogwoods are often grown for their winter stem color which is red or yellow. There are also varieties with attractive variegated leaves. zones: 3-7S / 3-8W
Cotoneaster, varied group of shrubs from small to tree like, white flowers, berries in autumn of various colors.
Lonicera nitida, golden leaved evergreen, good for hedging too.
Mahonia, especially x media
Pyracantha, spiky white spring flowered autumn berried extremely versatile shrub
Taxus cuspidata, Japanese Yew, for a very formal hedge or as a specimen, very hardy in cold areas.
Vitis coignetiae - Crimson glory vine One of my favourite climbers this one, large bright green heavily textured leaves up to 12" across whose autumn colors earn it its common name.
Fruit and Vegetables
Potatoes, the traditional crop to break up clay, they don't need a fine tilth, they can push their large strong roots and tubers through the soil and the act of earthing up helps break the soil up and suppress weeds. Main crop or late rather than early varieties as clay soils are slow to warm up in spring.
Legumes, Runner / French / Broad beans, Peas. When the crop is over cut the plants off at soil level, leave the roots to rot under ground so adding organic material and nitrogen from the root nodules that the plants form.
Brassicas, cabbages, brussels, kale (not cauliflowers though). Main crop or late rather than early varieties as clay soils are slow to warm up in spring.
Soft fruits, most types do well as long the soil is not waterlogged at any time.
Fruit trees, Apples, Pears and Plums in particular.
Root crops such as carrots and parsnips, they have difficulty in growing through the heavy soil and often end up with forked roots (unless you like the idea of growing "rude" carrots).
Any kind of early crops including salads, unless the soil has been warmed in advance by using a cloche, cold-frame or plastic tunnel.