After some exhaustive research, I was able to photograph the 30 plus mobs that can now be riden with the new saddle. This was not an easy task due to the fact that the saddle is unavailable in creative mode, so I had to craft all of them in story mode. And once you dismount your ride a saddle is taken from your inventory, so only crafting one will not get you very far. Each saddle contains 4 horse hides and one stick, so I had to kill a lot of baby horses for this experiment. :D
Another task that made this tricky, was that the hostile mobs have to be brought down to below 50% health before they can be ridden. And once you dismount they become hostile towards you again. One of the fun features about riding hostile mobs, is that they will still attack other mobs while you are riding them. This creates a great way to grind for gear. I was able to ride every type of giggle, but I only included the pics of me riding the Grampa versions. So you have to figure that there is 4 giggles I left out of the gallery.
Pets and non-hostile mobs
Pets can also be ridden. I was unable to test this on the pet spider due to my lack of materials, but you can ride a normal spider so I have to assume you can ride the pet one as well. I didn't test on the cat or dog either, because I was unable to ride mobs such as the bobcat so I didn't bother trying. You can look at most mobs and know whether or not if they can be ridden. But I did try to ride a squirrel and a lizard anyways. No luck
Let's just say, for scientific reasons, I did try to ride Paula. It was a no go, end of story. None of the NPC's are ride-able.
Under the sea Mobs
The smaller sea mobs, are unride-able, but the bigger ones can be ridden. If you submerge while riding your breath will still run out, so try not to drown.
I hope this answers a few questions, Enjoy.