I’ve always had a fascination for the “nose lines”, the lines that connect the lower part of the nose to the corners of the mouth. I simply find them attractive. Not all people have them, but for some they are quite apparent.
Leonard Cohen
Do they have a name? Perhaps something in latin?
Nasolabial fold/laugh lines/smile lines. 
CMC fnord!
I’ve always heard of the trough between the lines referred to as the Philtrum.
I think you’re talking about the good ole jowls. Prominent on such people like John McCain, Tricky Dick Nixon, JoeLieberman, etc
Not the same lines the OP is talking about.
When I lived in Ocala, Florida, in the mid-eighties, there was a local TV quiz show done live. Just a guy behind a desk asking questions and people calling in with answers. If they started to curse or say something inappropriate, he’d just hang up fast. I remember one night he got fed up with all the jokers and went on a rant, telling people they needed to behave and cut the crap, or else! Madder than I’d ever seen him. Then he returned to the question, which was (of course): What is the name for the lines which go from the nose to the corners of the mouth?
The phone rang, he picked it up, and the voice of a little kid quavered, “Booger valleys?”
That’s what I’ve called them ever since.
Ah, I see what he means now.
In their seminal work The Meaning of Liff Douglas Adams and John Lloyd (whose authority no one dares to question) call them “Des Moines”.
I’ve heard the term “frown lines.”
How do you frown? 
I’ve heard them referred to as ‘brackets’
No, des moines are the parallell lines from the septum to the top of the lip. I’m as certain of that as I can be without my copy of The Deeper Meaning of Liff at hand. (I can’t find it at the moment. Hope I haven’t lent it to someone, cause I haven’t a clue who that could be.)
ETA: Didn’t notice the thread had drifted to also consider des moines, but it’s still important to note that they’re not the lines in th OP.
I stand corrected. That will teach me to rely on my memory. :smack:
Nasolabial folds.
Correct me if I’m wrong, but it would seem that nasolabial folds, even if correct, is a very clinical term. When I google it, it’s mostly in the context of plastic surgery. Seemingly, some older people choose to fill their nasolabial folds with botox to appear younger.
Which brings me to another point. For some these folds do get more apparent with old age, like with crow’s feet. But then, you have people like Leonard Cohen, for whom they are apparent not only in young age, but also all the time, not only when he’s smiling or laughing. I don’t know. I find it interesting.
