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I always address my professors as "Professor [Last Name]" in an academic setting. However, in an non-academic setting such as running into them out and about, would it be considered rude to say "Hi [Last Name]!" to greet them?

Would it be better to address them by their first name even if you have never used it before, or always prefix their last name with Professor?

Would it be rude if I called you by your surname only?

It depends.

How well do I know you? What is our relationship? Is there a significant age gap between us? Are we in a fraternity or the military?

As a student, I never once called a professor by only their surname when speaking to their face, formally or otherwise. They were always older than me (even if just by a few years) and we never had such an informal relationship that surname alone seemed appropriate.

I did call some professors by just their given name. This was usually when they specifically requested it or when I knew them quite well. As I rose higher through the ranks in academia

(freshman -> sophomore -> ..... -> graduate student -> PhD candidate....)

calling professors by their given name became more common.

I'm sure there could be some professors that would be fine with being called by their surname alone. Most would not be, however. This is not unique to academia. Most people in their 40s do not like being called by just their surname by 18 year old kids. Even when I was a 25 year old graduate student, I'll admit that it would sort of bother me if students called me by only my surname (even outside of class). We weren't in the army or something. Just call me by my first name.

Obviously saying something like "I had combinatorics from Levenworth and topology from Kostanza" when speaking to fellow students is a different story. There's no need to worry about offending someone when referring to them by surname alone when they are not there.

In the U.S., in this year, at my age, I would be surprised if anyone addressed me by my surname. But that is a very different thing from reference to my work, where I'd mostly expect people to say "Garrett, [year]", or something similar. Still, in an in-house situation, to hear a speaker say "Paul's work..." would not be jarring, and might be more congenial than "Garrett's work...".

... although the times that a speaker has said "Garrett's work" and waved to me in the audience were perfectly fine.

I guess some nod of respect is the only substantive point, and this is dependant on the local culture... for which there is no clean algorithm.

As Vladhagen’s answer suggests, this question is highly dependent on the relationship you have with the professor.

As a general rule, I try to match the level of formality that I see from a professor. For example, if a professor signs emails to me with just their first name, then I respond using their first name in an email. If I see them in person at this point, I think it would be fine to use the first name (but I typically do not). When in doubt, always err on the side of formality.

If you have never used their first name (or if they have never indicated that using their first name in an academic setting is appropriate), then I would not recommend using it if you see them out and about. This could create a potentially uncomfortable dynamic, especially if other students do not refer to them by first name.

As a grad student, one of my professors has indicated that they are perfectly fine with students addressing them by first name only. I would personally prefer to be more formal, but since my relationship with this professor is very casual and friendly (and it would almost be weird to say “Dr. X” or “Professor X” at this point), I’ve adopted a middle ground of omitting the title and only calling them “last name” in both academic and non-academic situations. It is important to note, though, that they are a younger professor.

I wouldn’t find it rude, but I would find it weird. I can’t think of any situation where you wouldn’t be better off using a first name instead of a last name without a title. If you’re going to be formal do it right, and if you’re going to be informal use first name. We’re not on a football team or in the military.

(It’s of course totally fine and normal students talking to each other without the professor present, to just use last name and no title.)