Problem: Meshes appear "pinched", vertices go flying off in random directions, or some vertices do not animate

All of these problems are usually caused by missing vertex weights on skinned meshes.  Double check that all of the vertices in the skinned mesh are assigned to a vertex group and have a weight greater than 0.0 for at least one group to which they are assigned.  You can find out which vertices are missing a group assignment by switching into edit mode and using the "select" button in the "Vertex Groups" section of the "Link and materials" panel in the editing buttons:

Cycle through each vertex group and hit the select button. Once you've gone through all the groups, any vertices that remain un-selected are missing a group assignment.