RE:RE:RE:RE:drillingSporty, The most common that I've seen is because of stuck BHA. If they can't jar it free, then they find the "free point" where the drill string is not stuck. They do that by running a wireline inside the drill string and putting tension on the drill string. That's all that I know about that. A fishing expert can tell you how they find the free point of stretched drill string using a wire line tool.
After finding the free point they "shoot off" or part the drill string above the free point. Again, fishing expert can explain that. Explosives, I think. Then they run in with washover pipe to clean the annular space between the stuck pipe and bore hole. Then they run in with an "overshot" to latch on to the stuck pipe and pull it out of the hole.
If none of the above works, then they have to decide whether to keep trying or kiss off millions of dollars worth of BHA, run cement plugs on top of the stuck pipe and "side track" around it to drill new hole. This side track process takes a long time because it involves a lot of trips into and out of the hole and a lot of time consuming directional drilling.
But if there's a radioactive source like cesium in the BHA then different rules apply. I really know nothing about the rules but have been told that some governments would rather see a company go bankrupt spending money to get a radioactive source out of the hole rather than leave it in the hole.
So in such a case side tracking is not an option. They will eventually recover it but at what cost and in how much time? This could take a while yet. As for who pays it's usually the operator as long as the drilling contractor did nothing to cause the stuck pipe which I've never seen happen because they don't allow idiots near the rig floor much less set them up as drillers or toolpushers.