I know there are a lot of different ways to remove wallpaper, ranging from elbow grease to chemicals and steam machines.
What way(s) work(s) best?
It depends :)
It really depends on how old the wallpaper is, if there are multiple wallpapers glued over each other and how much glue was used.
If the wallpaper is not too old, and was put on in a proper way, usually, a steamer works very well. It has the advantage that it loosens the glue without making the wallpaper to fragile to pull it off.
I usually try to stay away from chemicals, because they always have some danger for health associated.
I've removed all the wallpaper from our house. In the three bedrooms, it was painted over. I think I've seen the gamut of wallpaper removal.
Easy Wallpaper just stripped off in nice sheets. This was in the dining room. It wasn't painted over. It took Maybe 15 minutes to do a 14' x 10' room.
Hard This is the technique we used for two of the bedrooms where the wallpaper had been painted over. Messier, much more work, but the only option.
Extreme The last bedroom we did, I came close to giving up and resorting to either tearing out the drywall or some other extreme measure. We tried everything, in the end it took 6 nights of work to do a small (10' x 10') bedroom. The paper was very, very stuck on and then painted over.
The steamer could not soften up the underlying glue layer because of the paint, even with very liberal perforation. If you held the steamer on long enough to get the paper off it had already softened up the underlying drywall and you'd very easily gouge the drywall and remove paint/backing.
So we resolved to first scraping off the top layer of paint exposing just the wallpaper backing and then using the normal technique for wallpaper removal with just the backing.
(I will have to check my notes, as I don't think this is quite the exact steps yet.)
A steamer is by far the best option: I've had great success even with multiple layers of woodchip. (Ugh!)
It can definitely help to rough the surface of the paper before steaming, especially if it's been painted.
If it's a real mess then it is actually feasible to plaster over paper, but I'd personally not recommend that approach unless you know a good plasterer or can do it yourself easily.
I've found you don't need a steamer or special chemicals. If you do use a steamer, be really careful not to damage the wall. Plaster will break apart, for example.
Instead, I've used one of those wallpaper scoring tools to cut small holes in the surface. Then, apply hot water from a spray bottle liberally over the paper and let sit for 30 minutes or so. It should then start peeling away easily with a scraper. Of course this depends on the type of glue used. If this doesn't cut it, you may need to mix some kind of glue solvent in the sprayer. This process worked on a circa 1950's house on plaster walls.
My wife and I just renovated a house that had 5 layers of wallpaper in the living room (including the "Hay" wallpaper from circa 1985) and we bought two steamers and went to town. It was the best solution we found. You don't have to spend a lot, just get the inexpensive plastic steamer unless you plan on doing it for a living.
She did re-use the steamers several more times for her painting business and they still work well.
Fill a plant sprayer with hot water and dishwashing detergent
and spray the wallpaper several times and let is soak for 30 minutes. Once the mixture has been incorporated well use a filling knife to remove the wall paper.
Repeat this process for wall paper pieces that stubbornly stick to the wall several times in order to remove it.
Note: this solution is applicable for removing wall paper from a plasterboard as well.