The much of this information was collected by Eben The Scrooge. Most credit goes to him.

Scribblenauts players are always looking for beasts that can help them fend off opponents, and you can only spawn God so many times before you want to try something new (not to mention you can't ride God). Luckily, you can tame almost any beast, and those that are rideable will gladly let you hop on their back, after you tame them properly.

Choose your Creature

While any Scribblenauts players have ridden some kind of creature or another, many users might not be aware of the sheer number of creatures you can ride. The best way to tell, is to try attaching a saddle to the creature. If you can put a Saddle on it, you can ride it. Some of the more unusual mounts include the Griffin, Cthulu, the Kraken, or a Sand Worm.

A Full Stomach Means A Happy Monster

One of the simplest ways to make your monster happier is to feed it till it falls asleep. You won't be able to simply toss Rats to a TRex though. The food needs to "look like food", meaning items like Steak, Meat, Chicken (Food), Drumsticks, etc. If you give them an animal, that when killed, turns into food, that will work as well. Once you have fed the monster three times, it should go to sleep, and when it wakes up, it will be much happier and allow you to mount it.

Put it to Sleep From a Distance

Just in case you need to get your monster to sleep faster, there is always the Tranquilizer Gun. This will allow you to keep your distance, and put it to sleep instantly. The Tranquilizer Gun will even work on creatures like Robosaur, despite it's metallic skin. By the way, if you want to wake a monster quickly, just tap it to interact with it, and it should wake up right away.

Weaken Your Opponent

Making a creature sick will actually tame it. To make it sick, the best option is to summon a radioactive element like Uranium or Ununbium and drop it on top of the creature. It will fall sick for a few seconds and then it will come back up tame and ready to ride! This is one of the best ways of taming a creature because you can do it from anywhere on the screen and you can use it as many times as you want.

Make Friends with a Bowshot or Mind Control Device

The quickest and most effective way to make a monster happy however, is with Cupid's Bow or Mind Control Device. The best thing to do if you want to tame it over long distances you should summon a Love Arrow and then drop it onto the creature. One shot (or drop), and the monster will instantly stop attacking and allow you to mount him. It should be noted that the Tranquilizer Gun, Mind Control Device, and Cupid Bow are one shot weapons, aim well.

Special Objects

There are a few exclusive objects in scribblenauts the will calm select objects. For example if you want to have a giant enemy crab guard you all you have to do is is summon a crab and hold it and it will fight to the death to protect you and the crab. A lion tamer will make a lion and a liger peaceful. You can turn anything into a skeleton warrior that fights for you using necronomicon (or book of the dead).
Let's Go For a Ride

Once your monster is tamed, you can simply tap it, choose the ride option, and go for a ride. Certain monsters work better underwater, on land, or in the air depending on it's domain. Also keep in mind that if you are wearing Wings, you can temporarily fly with whatever beast you have. Oddly enough, Cthulu can't fly even though he has Wings. Yet if you have Wings, he will not only fly, but his wings will flap and give you a little extra lift. You can also use your monster to attack any other object just by tapping it. Go out and conquer!