Like many animals, the only tame and domesticated animals are found in zoos or sanctuaries. Zebras are the same. Zebras have not been domesticated yet. Horses have been domesticated, but there are differences between zebras and horses. Animals are known to be considered wild animals and some of them are expected to live in the wild. This mainly includes zebras.
They can be domesticated and trained, but people don’t believe that they should do so. That is why they most likely have not been domesticated. They are not equipped to be ridden. It doesn’t seem like the zebra will ever be trained because people do not consider the zebra to be trainable animal. Also, it seems that zebras are slow and small so they may not be ideal for the reasons animals are trained.
Anyone's guess, I think, why zebras have not been domesticated but it may be due to more aggression in their make-up, more wariness over consorting with different species or lack of intelligence. It takes intelligence to learn, and domestication does involve learning new skills.
However, skilled safari guides have seen instances of zebra domestication but it takes breaking a zebra's spirit to achieve this. Many efforts were made to train zebras for riding, drawing, and carrying during the late 19th and early 20th century. But they haven't been bred like horses to serve man, so they have extremely weak hindquarters and backs as well as the aggressive spirit.
Many people have tried to domesticate zebras but their effort has not been fruitful. It is impossible to domesticate a zebra. You would expect that to domesticate zebras would be easy, since their close relatives, horses and donkeys have been domesticated.
Zebras are significantly smaller than horses; the discomfort of having a passager for any length of time is likely to activate their worst tendencies even when they have been trained.
Zebras are unpredictable; they can be aggressive and also attack when they sense something to be a predator. They can easily maim or kill a person. That’s the major reasons they cannot be domesticated.