First, a simple one. This is just a straight center post that would be bolted to the baseboard SECURELY and that would have support plates screwed onto the post along the height of it. I would use a threaded rod (yellow) and nuts with large washers (green) to support the cakes on the rod. The cakes would rest on the washers on top of a brace like this (click here). The brace would support the weight of each cake. I would make sure that this kind of thing was attached VERY securely to a wood base (use a wooden tabletop that you can find at the hardware store).
Another very simple one is a standing object that has a narrow base and is fatter on the top. Again, attach a threaded rod (blue) to the base securely with some type of flange system. The boards on this one (pink) would be resting centered on the pole with nuts and washers holding them up. The center pole should go up into the top tier of the cake, and there should be regular dowels (not shown) supporting the stacked tiers. I would make the handle and lower part of the brush under the bottom tier out of modeling chocolate.
Here's one that's trickier, but is still centered so it's balanced. The yellow shows the center pole that's attached to the board. I would put a PVC pipe over the center pole on the top so that I could insert copper tubing (blue) into the top to use for armature for the legs. Depending on how long the legs are, I would pack an armature wire around the copper tubing to bulk them out before covering them and the arms with modeling chocolate or fondant. The cake would be stacked like a normal stacked cake on top of the pink plates indicated in the photo, and there would be dowels in each tier to support the top tiers. I forgot to add a plate under the head but that would be cake too!
The last one would be the most complicated because of the double post in the center. Since the horse is leaning forward you'd need to do a post that went through and was used as one of the legs. Use modeling chocolate and build the leg into shape using the straight-centered pole as the very center, but kind of "bend" the chocolate around the pipe. The support will be straight but the leg will appear bent.
There should be another post that went through the other leg and is attached to a base that fits the shape of the horse's body where the bottom pink plate is indicated in the picture. That would provide a solid platform that the cake would sit on. You could build that beforehand and have it ready to stack the cake tiers. Build the horse's rear end out of styrofoam that you glue to the bottom of the base and cover the whole thing ahead of time, or wait to do it with the cake so that it matches better.
When you get to the part of the horse that starts leaning forward, you'll start stacking the tiers a little forward and add a second support pole toward the front of the horse. That will go straight up through the upper tiers and into the head of the horse. The pink boards show where the boards supporting the cake tiers will go, and those will be doweled like a regular tiered cake. The yellow lines are copper tubing that can be put into the cake to support the legs. I would probably put them UNDER the pink boards, not above them like int he photo, since that would anchor them more securely. I'd build them out with wire armature fabric to bulk them up, then I'd cover with modeling chocolate.
If any of that doesn't make any sense post a question. But making internal structures is just a matter of looking at the item and figuring out how to fight gravity to get the cake to stay up!
For some visual lessons on how to build internal structures, you can sign up for a Craftsy class or two to see some examples of structures being made. Some good ones are Topsy Turvy Cakes and Building Better Cakes (these are affiliate links).