I've had a habit of always starting with "Dear Prof. X" but have noticed professors sometimes drop the header past the first reply. For shorter emails, is it polite to omit these formalities from my end? Like "Dear Prof. X, Thanks for the information, I'll make sure to do that. Best, Me" seems a bit cumbersome.
An important thing to note here is that often these dropped headers don't happen because the email is merely short, but usually because the email chain starts to approximate a conversation, i.e. a quick succession of short replies. So the formalities shift from those of writing letters to those of talking in person. So maybe take some cues from there.
In particular, even if it is short, not starting the first message with some greeting would be a bit impolite. The same is true if some time has passed between messages. (You'd greet somebody if you meet them again, even if you just met them yesterday and didn't talk to anyone in between) But a quick ABAB-exchange within a few minutes doesn't really warrant repeated greetings.
And of course, as mentioned in the other answer, when in doubt, just err on the formal side.
In my experience, it should be fine to drop the salutation given that the other person has done so before you in that same chain. But if in doubt then err on the more formal side. It's always a good idea to emulate the email style of the other person in a one-on-one chain (unless you have some reason for wanting to maintain formality).
I usually find it's good to start and end formally, but for little in-between messages that are around one line and easy to answer it's fine to be informal. Unless the person is really arrogant, they won't find it disrespectful if you do this (especially if they've done it first), although it depends a bit on your relationship to them. Maybe something like this:
You: Dear Prof X, can you please tell me about whatever. Regards, Infinitus.
Prof: Dear Infinitus, here's some information, Signed, X
You: Is that A or B?
You: Thank you, I'll get on it. Regards, Infinitus.
Is it fine to exclude an email salutation for short messages?
In academia or not, a good reason to avoid short e-mails is to help the receiver know the beginning of a chain is legitimate.
Professor Newton, I'm having trouble with today's assignment [link to a "picture"].
This is does not provide the recipient enough info and has the appearance of some click-on-my-link exploit.
Professor Newton, In your lecture about apples falling from the tree to earth, I did not understand the shift to earth and sun. [link to a picture]. Student name
Certainly one should be careful before clicking any embedded link, yet providing relevant and timely info, along with what-ever courtesies you like, helps the Prof save time in deciding your e-mail's legitimacy.
(I'm in the US - I lived in Canada way back when I was first a grad student)
I don't inhabit the academy, but I did a graduate degree 5 years ago, and I did graduate work a long time ago (way before the advent of email).
I don't think I've ever used "Dear" in an email - ever, and I don't think I ever addressed anything to my research supervisor in the pre-email days with "Dear". I did always include a salutation along the lines of "Dr. Whatzit" or "Prof. Whosit" in the first email of a chain (or in a memo to my supervisor way back in the day). Like others have said, it goes away after the first use.
I never expected a salutation in return. I have no idea if I ever got one. As you may have figured out, I was older than most of my profs (all but one) when I did the degree a few years ago.
I see a lot of answers supporting dropping salutations once a particular email chain gets going. But I also don't use them with anybody who I interact with a lot. This includes family members and friends, but also my boss and close colleagues. When I was a grad student it included my advisor and professors that I was in a small class with. But when I email someone who rarely hears from me, I'll start with a "Hi [Name]" or something. When I was a grad student emailing a prof at another university to ask about one of their papers, I would start "Dear Prof. [LastName]," just to be especially deferential since I was a total stranger. I think a lot of people operate this way.