Sprains are one of the most painful things ever - you get a sprain and you know about it for days afterwards. By definition, a sprain is an over-stretching or wrenching of the ligaments of a joint without actually breaking or dislocating the joint. The most severe kind of sprain is where the ligament actually tears away from the joint. Ouch! (I'm wincing just thinking about it!)
I have not encountered very bad sprains, and am not sure that they can be treated at home. A serious sprain as described above can require surgery, so if you suspect you have done damage like that, please consult a doctor. Most people, however only experience sprains that keep them laid up on the couch for a week with swelling and bruising. They are very painful however and quite annoying to the modern, busy, 21st century person. :)
Recently, my brother (9) sprained his ankle when he hit it up against a brick wall while playing cricket in the backyard. He had a lump just above the ankle bone and was yelling when I touched it, so I knew he was in a fair amount of pain. He couldn't walk: he just lay on the couch and looked miserable. I decided to try a 'new' remedy that I hadn't done before. I soaked cotton balls in apple cider vinegar, placed the sopping wads right against the ankle and bandaged it up. Brother9 loved it - I think he felt like a war veteran. :) He slept that night with the bandage still on. The next morning there was still some bruising, but Brother9 was running around like usual. He said there was no more pain and though I made sure he took it pretty easy all day to make sure the ankle healed completely, he had absolutely no more trouble with it.
I was rather pleased with the results so, when my paternal grandparents were visiting, I decided to try again - on my Grandpa. Grandpa enjoys trying new natural remedies, and willing specimens are so much easier to work with than those who doubt your capabilities. Grandpa has a bad knee that has bothered him for years, but while he was visiting, he tweaked it somehow and was limping around the house with it all strapped up with electro-magnetic bits applied. By the second day, he wasn't much better and was struggling to move around properly. He readily agreed to let me have a go at it. I got out the first aid kit and soaked a dressing pad in apple cider vinegar. We applied it to where Grandpa's knee hurt most and bandaged it up. Again, we left it overnight. The next morning, the bandage was off and Grandpa was moving around like normal. He then went on to spend most of yesterday kneeling on the concrete painting the fence with hardly any trouble at all. They went home today, Grandma getting all the particulars from me on how to apply the vinegar so that she can continue the "treatment" at home.
Once again, if your sprain is really bad, get it checked out. Most sprains, however, can be treated at home at the tiniest fraction of cost that you would spend at the doctor/x-ray/hospital etc. :)