Burned Alive
In the beginning of the movie jack sticks a flaming torch in his mouth? The fire covered his entire body pretty fast, even for a body made of straw (he was in his scarecrow form). When you burn a human body for too long all that is left are the bones. Everything else is gone.
Dismemberment
This one might be a bit self-explanatory. She can fall from high places and break into pieces(become dismembered). She can then sew herself back together and will be completely fine after.
Skinned
Oogie Boogie's body is sewn together around the sides or edges. His skin is instead burlap, and his innards are insects. Insects are nature's cleaners. He could have been skinned and then left in the ground? His innards may be the very swarms of insects that devoured his skinned corpse. So in death they had to come together to allow his body to be whole. Being without skin he needed something to hold it all together, hence the tightly sewn burlap like skin.
Poisened
Sally has been mentioned to have poisoned Dr.Finkelstein on several occasions. With him already being deceased being poisoned wouldn’t really affect him much. If it were to be what had killed him, this would explain why he’s affected, meaning when he does get poisoned AGAIN in death it only knocks him out and gives him a severe headache.
Frozen
It is believed that his cause of death was because he was frozen, mainly due to his lips. When people are frozen alive their lips turn blue and their face turns pale or a white-ish hue. He also chose his costume to be a devil. Something that revolves around heat, showing a strong desire to be warm. Now could that have been a coincedence? ... Maybe.
Suicide
Shock's lips are pale and have a purple-ish tint to them. This can happen after you have passed from any cause of death aside from three things: Burning, freezing, and drowning. Which could mean that she passed away from a natural death (which is unlikey because she is a child) or she took her own life. Shock shows a desire to go against any kind of intimidating/scary figure. However, she doesn’t seem to mind Oogie Boogie for she is one of his "henchmen". Could they have known each other in life?
Drowned
Barrel's lips are of a blue-ish/purple-ish hue just like Lock's and Shock's lips. When drowned, your hair has a slightly sea weed look to it. Barrel's hair definitely had a sea weed resemblance. Your body also doesn’t decompose the same way in water. The skin tends to cling to the body, and Barrel chose his costume to be a skeleton. He could have possibly wanted his body to stay together (attached to his skeletion).
Broken Neck
It's believed that the Mayor is/was bi-polar. In the movie you can see that his mood changes a few times. He has two faces, one being happy and the other being sad. His neck could have gotten broken due to the fact that he is bi-polar and has episodes of sadness and/or anger. When his mood does decide to change his neck "breaks" and twists a full 180°.
Electrocuted
It's speculated that Zero was electrocuted because the last thing to leave the body when electrocuted is smoke and Zero’s body looks a lot like smoke when he's moving around or floating around. He could have been Jack’s dog in life as well as in death. Lastly, Zero's nose is a lightblub. He can use his nose to light up areas that he is in, meaning he has to have electricity coarsing through his body for his nose to be able to light up the way it does.