Does anyone say “corner of the lip”?

From my experience, no. However, if you said it I would still know exactly what you’re talking about.

Can't answer without context. If I have a blister on the corner of my lip, then yes, I'll probably say "corner of my lip".

Depends on how technical you want to be. Could say corner of the upper lip. Right corner of the lower lip. Or corner of the lip. Etc. All would be understood perfectly.

Unless you were specifically referring to the corner of one lip for some reason, no, you wouldn’t say “corner of the lip”. It sounds strange.

Not that I’ve heard

No I've never heard that in my life lol It's hard to describe but the mouth is the whole shape, it has corners since it's both lips. A corner is formed where two lines meet. A lip is more or less just a line, a simple shape. We say something similar with eyes, the eyelids are "lines" and the eye is the whole shape. The eye has a corner ("out of the corner of my eye" is a commom phrase), the eyelids do not You'd just say the side or edge of the lip

Only 1 instance I know of, and it’s from “Elevation” by U2

Nope. Lips are lines, they don't have corners. Mouths come together, they have corners.

I like using a corpus (a big collection of texts and/or speech) for things like this because the responses here are still a small sample size of speakers. The corpus of contemporary American English gives 412 occurrences of “corner of his mouth” and 26 occurrences for “corner of his lips”. So mouth is roughly 16 times more common than lips in that set phrase. And if you click in “corner of his lips” was mostly used to describe smiles in fanfics and other stories. Not necessarily normal, conversational English. So it does seem like something that does occur but I would personally never use it, preferring “corner of ones mouth”.

No

Look at what a corner is. It's formed by two vertices, and in the corner of the mouth example, those two vertices are two lips. Where else do we talk about corners? The corner of 1st Street and Maple Avenue? What is the corner of 1st Street? It makes no sense, just like the corner of one lip makes no sense.

No

No. Because a lip is — for lack of a better description — more of a “line” than a mouth is.

I’ve never heard anyone say “corner of the lip” but plenty say “corner of the mouth”

No, I’ve never heard that and if I did I wouldn’t know what they meant.

No. I’m not saying it’s NEVER been used before but it doesn’t sound natural.

I guess it depends but in general, no. A lip doesn’t intersect itself. (In reality of course it has a boundary and could be measured). However, when we are talking about the corner of the mouth, we mean where the lips join. We are usually speaking to someone or about someone. You could say: Your upper lip has something on the right corner. You have some mustard on the left corner of your lower lip. The corners of her mouth turn up. He talks out of the corner of his mouth. (the side)

I’d like to know as well

No, I would always use corner of the mouth and that’s all I’ve ever heard be used as well. I would still understand what you mean if you used corner of the lip but it doesn’t sound very natural in my opinion.

No, “corner of the mouth” is basically a set phrase, and is used for the particular part of the mouth where the lips join at the sides. You might say something like “you have a bit of chocolate on your lip” to describe a situation where the chocolate is on only one lip, but not at the place where the lips join to form the corner of the mouth.