Had all three.
Broken bones can take months to heal, and often lead to arthritis in the affected area, as I can personally atest to. Bones shattered by gunfire are painful for a long time, like decades.
Now a really bad burn, like over a very large portion of your body, would be worse over the short term, but most "normal" burns will heal much more quickly than broken bones, though the pain from burns will definitely be more intense for the short term. I got burned over a large portion of my chest once (never fry anything while not wearing a shirt) and was in intense, "take your breath away" pain for three days; I could barely move.
You also forgot crushing injuries. I am coming off a really bad case of "M1 thumb." It got infected, too, despite the fact that it was a closed injury and the pain was bad enough that I couldn't sleep for two nights.