There has been another thread on this topic just a few days ago, so check out the answers there.
There are two main type of thistles which are a nuisance in pasture - creeping thistle and spear thistle, and they need different treatment.
Creeping thistle can be eliminated by regular cutting, best with a lawnmower which collects the clippings. Eventually (a few years) the creeping thistle will have lost any energy it ever had and will stop growing.
Spear thistle can be dug out by hand but timing is important. Wait until a flower shoot has come up, but the first bud is just short of opening (usually May-July), then dig out, tap root and all. The plant has then invested all its energy in its flower stalk so is less likely to regrow if you have accidentally missed a bit of the root. Depending on your acreage, clearing spear thistle by hand can seem daunting, especially as they do reappear next year, but we have virtually eliminated them on our land, without a single drop of weedkiller.
Spear thistles are mainly biennials so spread by seed, whereas creeping thistles are perennials, which spread by roots.
Spear thistles have drawn up plenty of nutrients from deep down, so are very nutritious and would be a useful source of compost, if you could get the heap hot enough to kill any seeds. We usually burn ours but we are aware that it's a huge waste of a great resource.