Are you talking about flying rigs, to hold them up while they are jumping in the air?
I don't use action figures, so some things might be different, but the 2 ways I know of to support a puppet in the air are hanging from thread or fishing line, which is almost (but not quite) invisible, or a more solid support which holds the puppet more firmly, but has to be removed in post production.
I use a ball and socket rig for heavier puppets, and several strands of 3mm armature wire twisted together for the lighter ones. (I know, you didn't want to use armature wire.) Usually I sharpen a point onto 2 of the wires, and poke them into my foam puppets on the side away from camera. For hard plastic figures I guess you would need to drill a couple of holes. Having 2 stops the figure spinning around like it could on only one wire.
Then, because the rig is visible, I have to erase it in a program like Photoshop or After Effects. (I happen to use TV Paint Animation for this, but PS Elements could do it, or one of the free open source programs I think.)
When my puppets are standing on the ground I use tie downs, bolts that go up through the set floor and screw in to the puppet feet. I don't know how much you would have to modify the action figures to do this, but one way is to fix a nut in the foot with epoxy putty around it to hold it in place.