Imagining, for a second, that dinosaurs still exist. In this world, for whatever reason, people domesticated dinosaurs and- much like pet goats and pigs- they eventually found their way into our homes. Could dinosaur species replace greyhounds and horses for racing? How would the costs of their maintenance compare to cats and dogs?
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Since birds are dinosaurs, perhaps we can draw some analogies from there.
Small birds would be good analogues for very small theropods, so most small dinosaurs could be expected to be very quick moving and exhibit territorial and mating behaviours and displays that humans might find fascinating to watch. Most examples are now thought to have been covered in proto-feathers, so colourful displays by the males seem pretty much assured. I am not clear if dinosaurs would be able to "sing" the way birds do, so perhaps you might have to put up with shrieking noises instead.
Larger theropods might not make good pets, since in the fossil record it seems clear that theropods were carnivourous. An Emu or Ostrich sized Velociraptor would need some very extensive kenneling and you would want to be quite careful at feeding time...
Of course the most famous theropods are members of the Tyrannosaur family, and even lesser subspecies would be pretty much impossible to manage. You would have to live on a ranch to supply the food and open space the beast would need to hunt, and the neighbours would have a few issues with your "free range" Tyrannosaur, regardless of how good it tasted on the BBQ later on...