Ostrich rides are a common attraction in South Africa, but that doesn't make them a great idea. Ostriches are adorably odd little creatures, and ostrich farms can be a great way to interact with these birds. But their awkward bodies weren't really made to handle the full weight of a human being, and doing it for a career isn't really a great life for the little guys.
Elephants have been trained by humans for thousands of years. Their enormous size and strength have made them useful for everything from pulling wagons to warfare. However, as information about their social structure and intelligence emerges, the abuses commonly used to "train" these animals (especially in Thailand) become harder and harder to stomach. Today, there's much debate about whether it's ever okay to ride an elephant, but at the very least, look for sanctuaries and rehabs that care for them.
Reindeer (aka caribou) are pretty widespread in mountainous regions all across earth, from Canada to Siberia. They've been domesticated since the Bronze or Iron Ages, and they've been toting people around for a long time. Usually, when you see reindeer, they're pulling a sled, Santa-style, but the larger Siberian ones can be ridden, and several groups of Mongolian people ride them, as well.
Hippos are as famously as mean as they are adorable. They mark their territory by pooping and using their tails to fling it everywhere. Their cute little "yawns" are a threat display, and if you ignore them, they will still charge at you with murder in their eyes. There is one instance of a farmer, Marius Els, who raised a hippo from five months old and was known to ride it and treat it like a pet, uploading videos about how much he loved "Humphrey." But six years later, Humphrey apparently had enough, chewed Els up, and drowned what was left.
(image via merciplow)
Camels are amazing. From their ability to withstand incredible changes in temperature to the legendary amount of time they can go without drinking water, these animals are singularly adapted to surviving in the desert. And since camel saddles have been around since at least 1200 BC, these animals are pretty used to being ridden. They're everywhere in petting zoos, but camels are most commonly found in northern Africa and the Middle East.
"Hop On" (kids only)
Most commonly found in Peru and other parts of South America, llamas have been pack animals for a long time. That said, they're not used to the intense weight of an adult person. If you have a child who weighs less than 50 pounds or so and you use a saddle to help distribute the weight, llama riding can be safe and fun for a child. Just remember to take it easy and take care of the llama. (It's still a better idea than riding zebras, who are not only small but also mean enough that they have defied centuries of attempts to tame them.)
Water buffalo are common in India, the Philippines, and in other places across Asia. More human beings depend on them than on any other domesticated animal in the world. They're used to till rice fields, as well for their meat and other physical products. Since they're used to hard labor, they're often ridden, and even raced. You can even find people sledding behind them on occasion.
Dolphinariums and other places where humans interact with dolphins are pretty controversial. Dolphins are clearly intelligent creatures, and there are enough ethical issues surrounding captive dolphins and "swimming with the dolphins" programs that several countries have banned them outright. One thing that is pretty clear is that grabbing onto a dolphin's fin while riding isn't great for them. Maybe if you're stranded in the ocean and one appears to offer help, then go for it. But bear in mind, it's just as likely to attack you for fun. Dolphins can be schmucks.
When Charles Darwin first encountered giant tortoises on the Galapagos, his first instinct was to climb on top of it in an attempt to ride it. Others have tried since, and while it's not hard to find videos online of children sauntering along on top of them, it's horrible for the gentle creatures. A Chinese zookeeper was fired for poking tortoises to move in order to give children rides, which honestly seems fair. That's a terrible idea.