Usually, the ideal voice is a clear pitch. I will use "gristle" as a word for all of the reasons why a voice isn't clear. For example, a "gravelly" voice is not clear. Similarly, if you have phlegm in your throad, your voice won't be clear.
Usually, gristle is thought to be a flaw in singing. Usually it is. But sometimes singers intentionally include gristle, for effect. A classic example is Barry McGuire's "Eve of Descruction". McGuire sang a vocal version for musicians to follow. It was late, he had been singing all day, and his voice was ragged. The plan was to put in a better version later. However, the ragged version sounded so good, they kept that. I cannot imagine the song being as good without the ragged edge.
Of course, that does not answer why a ragged voice is desirable, or why gristle would be added to any song.
As near as I can tell, when people are sexually aroused, they get some mucus or saliva in their throat. This adds gristle.
It is a distinctive gristle. It is possible that you could add, for example, a creak to a voice and that would give it a feel of arousal. But it is unlikely. Essentially, the different types of gravel have different, recognizable sounds.
It is subtle, but you can find it on the internet here. In a wonderful but obscure CD, Aubrey Lande adds a hint of gristle to the song "Loco-Motion". More mainstream, it seems that Britney Spears has no trouble mastering this technique.
Also, a "dirty sax" has gristle. (The above CD also has a mildly dirty sax at the end.)
When you are in grief, muscles tighten in your throat. I have read that this is an involuntary contraction, and that you cannot do it intentionally. That is my experience -- I can probably mimic it, but I can't really do it. I think I did see an actor do it once, though. I think it feels like a "lump in your throat". In lesser form, you experience it as a "catch" in your voice. Vocally, the constriction produces a raise in pitch and some gristle.
This occurs relatively often in singing, but I have no examples for now. Essentially, any sad song can be improved with this technique. See Presley
Age, Size, Wisdom, Grit
There is something called a "creak". As people age, their creak increases. So you can, perhaps, signal age or wisdom by adding a creak. As noted, Karen Carpenter has a creak that to me conveys a mild image of wisdom. (Though it is more likely that she simply has a creak when she sings softly.)
If the creak becomes too large, then it is called "gravel". I am guessing that you need large vocal cords to produce gravel, so gravel can convey size and strength.
And when you are tired, your voice can get a "ragged" sound. I guess this projects a "tough guy" image.
I define a falsetto breakas a quick flash of falsetto. The falsetto break usually occurs on the start of the note, though it can occur elsewhere. Obviously, falsetto breaks are more suited to male singers, but I think maybe Karen Carpenter has a falsetto break on one of her very early recordings ("California Dreamin'").
Falsetto breaks are obvious in hill-billy music, where they are a distinctive part of the style. In pop/rock, they are not obvious. You have to know what you are listening for, and even then many falsetto breaks are hard to hear. They also probably occur more often than you think.
Anyone, even me, can produce the distinct falsetto breaks in hill-billy music. I cannot produce the subtle falsetto breaks that Elvis Presley produces -- my voice is simply not athletic enough.
I do not know the function of the falsetto break. Buddy Holly used them. I am willing to believe that he was just experimenting or trying to be different (or just showing off that he could do them). However, I find no sign in any of Presleys singing that he ever tried to be different or experiment or show off. Instead, as far as I can tell, he always tried to sing the song as he thought it should be sung.
Similarly, Gurfunkle (of Simon & Garfunkle) had a wonderfully clear voice yet he does falsetto breaks. He apparently was a perfectionist, so it is very unlikely that the falsetto breaks were an accident or unintentional. They are far too subtle to be showing off or experimenting or trying to sound different.
In theory, the falsetto break could be adding a harmonic element. But I don't hear it, it is difficult for me to believe that the singer could control the pitch of the falsetto break well enough to do that, and high pitches are not well-suited to harmony.
So, like I said, it's a mystery. Do they have function? It is difficult for me to believe that these artists include the falsetto break without any reason. But I can't say that there singing would sound worse without the falsetto break, and I can't see any additional meaning or feeling produced by the falsetto break.