I wonder if it's possible to pet a domestic crow, or an owl. Do birds respond to grooming as cats or dogs, for example? If so, then how does one pet a bird?
Birds do groom, but not like mammals do. People tend to call their grooming behavior preening. Preening removes dirt and parasites, arranges the feathers nicely, and distributes oils over the feather (very important for waterfowl.)
To answer your question better, we need to look at the specific species. Lots of birds preen each other socially, so if you wanted to pet a bird (in a way it likes) it's good to know how the species behaves.
You mention crows and owls. Crows are social so they may preen each other; they're also sophisticated (cultural) so it's hard to generalize. A 'bird nerd' on YouTube shows convincing footage of such behavior. I think family groups of crows preen each other, but I don't have a source.
Owls tend to be solitary and would not understand preening behavior from a human. A social species of owl might preen; I don't know. Of course, if you just go ahead and pet your owl like a dog, it may appear to like it. It's hard to say if it does like it...it's easy to anthropomorphize a bird with binocular vision!
I know first hand that Diamond, a tame Pionus chalcopterus, will sometimes respond to having his head and neck scratched in an interesting manner: he may make "panting" or "huffing" noises unique to this activity, lean his forehead or beak against my hand, and half-close his eyes.
I'll add to Ryan's answer by pointing out another use of preening: peeling the cover from newly-grown pin feathers.
How to (safely) pet
It's important to read body language correctly. Just as in Manderan it's funny to call your mother-in-law a mountain horse, you must learn the distinction between "scratch my neck" and "stay back or I'll bite". The ruff is raised normally as a signal of aggression.
Bring your hand up to his level wrist up and fingers trailing down in a relaxed posture. Make sure he sees it, without startling him. Human hands are not "natural" to non-domesticated species, so give him a few seconds to process. Higher thinking is slow and an optional "mode" so to speak. If he wants to be petted, he'll engage or remain neutral and undisturbed.
Diamond has learned through consistancy that a natural but superficial agressive signal is understood as "give me space", so a horizontal body position with beak in-line like an arrow (the "T-Rex position") and a slight (symbolic) darting motion means you should withdraw. But he may change his mind in a few minutes now that the subject has been raised.
The general points are:
- understand his body language
- be consistent in your own actions and associated meanings. They're learning a pidgeon with you (pun intended), not like dogs that intuitively know humans,
- give hime time to "think".
- develop a routine