I would say to keep your eyes open in early spring and autumn thats whanmost commercial clearance will be done.
Sycamore is agood woodto carve (about all it is good for, but thats another whinge), birch is easy - lime is nice.
If you are a beginer i would really advocate trying to work green wood as it is doo much easier - but you have to think about spliting the wood down to stop it cracking later.
Almost any green wood will be easier to work than (almost) any dried wood. Willow can be hard to split out as its fibers are interlinked somewhat.
If you volunteer for a locat wildlife trust of BTCV for a day when they are doing tree clearance and woodland management you can vertually guaruntee something usefull.
For smaller projects like spoons you can use quite small limbs - it does rather depend on what you wish to make?