Dinosaurs probably did sleep, though it’s all speculation, because scientists don’t know enough about their physiology.
The speculations are based on their structures and diets and on the habits of the presumably most similar living species, like crocodiles and birds, which do sleep.
Carnivorous dinosaurs like Tyrannosaurus rex and Deinonychus would probably have slept a lot, based on the analogy of predaceous animals of today, which, whether warm or cold-blooded, get their meals in big chunks after a quick pursuit, then go back and rest.
Lizards often rest after dining on mice, and lions and cheetahs spend most of the time lounging around between hunts.
Herbivorous dinosaurs probably did not get a chance to sleep as much, because they would have spent more time eating continuously throughout the day.
As for their sleeping positions, scientists suspect that two-legged dinosaurs would have slept lying down.
Sauropods like the brontosaurus, or apatosaurus, with their long necks and tails and their elephant-like stance on four straight legs, would have been the best candidates for sleeping standing up.
It is hard to imagine that Tyrannosaurus rex could get much sleep standing up.