The restrictions on 1st person (singular or plural) does depend on:

The genre of writing
The style guide being followed
The opinion of the teacher/instructor/advisor/publisher/editor
The context within the paper

Genre of Writing
Generally speaking, it is research writing that tends to favor not using 1st person. The reason for this is that research ought to be objective, not subjective, and too many inexperienced students (i.e. college undergraduates or younger, though even some graduates) doing research writing tend to write opinion, rather than properly researched and documented facts. So for these types of students, it is often safest for a teacher to say avoid 1st person. This helps force the student not to be thinking about what they "believe," but rather what they can "prove" via their research.
Style Guides
Style guides vary on what is allowed, but generally have some similar ideas. For instance, The Concise Rules of APA, 6th edition (2010) states for an appropriate use of 1st person (bold added):

Innappropriately or illogically attributing action in an effort to be objective can be misleading. ... To avoid ambiguity, use a personal pronoun rather than the third person when describing steps taken in your experiment.  Correct:    We reviewed the literature.  Incorrect:    The authors reviewed the literature. (sec. 1.09, pp. 19-20)

But then further clarifies use (bold added):

For clarity, restrict your use of we to refer only to yourself and your coauthors (use I if you are the sole author of the paper). Broader uses of we may leave your readers wondering to whom you are referring. (sec. 1.09, p. 20)

So in short, don't include in a "we" (1) the readers, (2) other people in the field of study who did not author the paper with you.
Similarly, A Manual for Writers of Research Papers, Theses, and Dissertations, 9th edition (2018), which tends to follow The Chicago Manual of Style in a lot of respects, says the following about avoiding 1st person (bold added):

Most instructors and editors do agree that two uses of I should be avoided:

Insecure writers begin too many sentences with I think or I believe (or their equivalent, In my opinion). Readers assume that you think and believe what you write, so you don't have to say you do.
Inexperienced writers too often narrate their research: First I consulted ..., then I examined ..., and so on. Reader care less about the story of your research than about its results. (sec. 11.1.7, p. 120)

And then later notes two more with respect to the 1st person plural as well:

But many instructors and editors object to other uses of we:

the royal we used to refer reflexively to the writer
the all-purpose we that refers to people in general (sec. 11.1.7, p. 121)

Between the above inappropriate uses, the manual notes some positive uses:

The first person is appropriate on two occasions ...

An occasional introductory I (or we) believe can soften the dogmatic edge of a statement ... The trick is not to hedge so often that you sound uncertain or so rarely that you sound smug.
... when it's the subject of a verb naming an action unique to you as the writer of your argument. Verbs referring to such actions typically appear in introductions (I will show/argue/prove/claim that X) and in conclusions (I have demonstrated/concluded that Y). Since only you can show, prove, or claim what's in your argument, only you can say so with I. ... On the other hand, researchers rarely use the first person for an action that others must repeat to replicate their research. Those words include divide, measure, weigh, examine, and so on. ... Those same principles apply to we, if you're one of two or more authors. (sec. 11.1.7, pp. 120-121)

Authority's Opinion
Style guides give some "allowance," but whoever an author is writing for has ultimate say on what precisely is allowed. This is generally noted in the style guides as well.
For instance, the second bullet point in the preceding quote about "appropriate" uses would not be a valid use by a number of people in authority who believe such metadiscourse (or signposting) that gives a discussion of what the author "will do" in the paper is bad writing for a couple of reasons: (1) regarding whether an author "proves" the point or not is up to the reader, whether the author believes it was proven or not; (2) what the author is showing, arguing, claiming, etc., should be evident within good writing without it being explicitly stated.
The point, however, is that the one the author is writing for (whether a class or a publication) determines the appropriate uses of 1st person; the author just needs to follow the rules laid down.
Context within the Paper
This relates directly to the caveats noted in the style guide section above, and also relates to the authority's acceptance or not. Some places in a paper may be appropriate to use 1st person, other places not. If an author is giving a personal illustration, it would be appropriate; but in a research paper, a personal illustration is generally not often used (since it cannot be verified; it is subjective). So whether or not it is allowed in a particular place will no doubt depend much on the context within the paper itself.

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