If an amateur filmmaker wants to use a short clip of a popular song, sung (covered) by a singer other than the one who wrote and popularized it, what permissions, if any, are needed?
That depends a good deal on just how much of the song will be used. Please read the answer to this question on this site.
The question is whether the use of the song except would constitute "fair use". Several questions need to be asked:
Would the use of the song in the film make it less likely to be purchased? That is, would someone who watched the film have 'satisfied" any desire for the song, and not be as willing to pay money for it? If so that weighs against fair use. If not it weighs for it.
How much of the song is used? the closer to the whole song, or the "essential core" is used, that weighs against fair use. If a song consisits of a verse plus a chorus repeated several times, and the film uses the verse plus one instance of the chorus, then the essential core of the song has pretty clearly been used. In the well-known Harper vs Nation case, a quote of 300 words was held to be the "essential core" of a book-length biography of President George H. W. Bush, and to be too much to count as fair use.
What is the purpose of the use? is it trasformative? That is, is the film using the song in pretty much the same way that the singer would have? or is it there as an example of its genre, or perhaps it is used satirically. The more different the purpose from the original, the stronger the case for fair use.
The character of the work being used. Since a song is a creative work, rather than a factual work like a news article, its protection is stronger, and the case for fair use weaker.
Note that fair use is always a case-by-case, fact driven decision, and no one can say for sure if a particular use is a fair use unless and until a court has ruled on it. It is also a very specifically US legal concept, and does not apply elsewhere.
Assuming that the use does not qualify as a fair use, permission would be needed for the copyright holder on the cover recording, and from the copyright holder on the song itself. The copyright on the cover recording might well be held by the cover singer, or by the (cover) recording company. The copyright on the song might well be held by the creator (shared by composer and lyrics authors if they are not the same person), or by the recording company that the creator has signed with. Or either copyright might be held by some other person or firm, as copyrights can be sold or assigned.
The film maker would need to contact these copyright holders and ask for permission to use the except. The holders might say yes or no, or ask for a fee. If there is no answer, that must be treated as a "NO".
If the film maker uses the song without getting permission, s/he could be sued for copyright infringement by the copyright holder, and possibly subjected to a court judgement for damages. This is true even if no money is made from the film, although damages will probably be larger is significant money is made. If the copyright holder does not choose to sue, nothing will happen.
EDIT: As a comment by @Dale M points out, the US has "Mandatory licensing" for music. This is provided under Section 115 of the Copyright law (17 USC 115). This means that if someone wants to record and distribute a cover version of a song, this can be done without the permission of the copyright holder. Instead, a "Notice of Intention" (NOI) is sent to the copyright holder and filed with the copyright office, describing what music is to be used and how. Then royalties are paid at a statutory rate (which may be higher than the current market rate). This process is described in Copyright Office circular 73 and in "Learn What a Compulsory License Is in Music" from The Balance.
However, a compulsory license is not available for the kind of use described in the question. A compulsory license permits making a new recording of a new performance of an existing non-dramatic musical composition. It does not permit distributing an existing recording without permission. And a compulsory license does not permit use of a musical work as part of an audio-visual work, such as a film or video.
A phonorecord does not include sounds accompanying a motion picture or other audiovisual work.
Section 115 does not cover sound recordings. Rather, it covers the reproduction and distribution of nondramatic musical compositions.
You cannot engage in the following activities with the compulsory license. You will need to seek permission from the copyright owner to: ... Make, reproduce, or distribute a sound recording publicly distributed in phonorecords
(From Copyright office circular 73, linked above)
So to use a published recording of a cover version of a publisahed song in a film, permission would need to be obtained from both the copyright holder of the recording (usually the performer), and from the copyright holder of the song itself (the creator or whoever the copyright might have been transferred to).
The general process of obtaining such permisison, not specifically for music, is described in "The Basics of Getting Permission" from Stanford University.
Also, US compulsory licenses apply only to music of US origin. They do not apply to music originally published outside the US. There are other limitations as well.