What can I plant?

Very tough indeed and you have two options, you either dig through the majority of the clay removing alot and replacing it with better quality soil like leaf mulch or you can look at the RHS's list of plants for wet soil, and because I'm so kind ill list you the latin names of the ones it has listed under shrubs

Aronia Arbutifolia
Calycanthus Floridus
Clethra Alnifolia
Cornus Alba 'Sibirica'
Kalmia Latifolia
Ledum Groenlandicum
Magnolia Virginiana
Salix Caprea
Salix Purpurea
Sambucus Racemosa
Tetrapanax Papyrifer
Viburnum Lentago
Viburnum Opulus and cvs

It doesn't say what other conditions will be required so research their frost hardiness and google away!!!

Good Luck...

Hi, I would look around in other peoples gardens near you to see what is happy in the area. Ten to one they will have similar soil conditions.
If you dig in the patches you want to plant and remove or breakup the clay on a dry day ,add some compost and add bought conditioning soil packs like athur bowers or some such not expensive stuff, or see if there is a local building site that has removed top soil and has some to spare. .Clay is very good when it has been broken up and dug in with better loams.
You need to get the clay broken up if you have solid layers...other wise things will struggle and waterlogging can happen. Hope this helps.. do it in little patches when you are plantig a new plant or two then you dont get back ache!!! and gradually you soil will improve. Recycle the clay by breaking it up and add to compost heap!

Hi its me again ... I have read your question again and now try the lawn care bit!
Fork the lawn to aerate. Do not do this too early in the season but April would be ok. Add some course sand as a dressing.
Other ways are to lift the turf bit by bit and remove some of the clay ,scratch the surface well and add a better loam. replace turf and tamp down. Try an area you think is the worst and see if it works. Also you could try going a bit deeper if the clay is thick to aerate and condition what is under your lawn.