Underneath, to me, connotes an aspect of being hidden.
"There were old grapes underneath the fridge" (= they were under the fridge but we could not see them/we did not know they were there [initially or at some point in time]); versus
"The linoleum under the fridge needs to be replaced" (= its position is lower than the fridge, but its existence is immediately known to the speaker).
"There were centipedes underneath the large rock we removed" (= the centipedes had been hidden; the speaker had not known they were under the rock until the rock was moved) versus
"The soil below rocks retains its moisture much longer than soil exposed to the sun." (= the speaker is already aware of the soil lower than the rocks; it's merely a statement of physical position).
So for me underneath connotes something usually physical, tangible, that is or was initially unknown or hidden versus under which is more neutral.
Beneath as discussed in earlier entries can for sure give the feeling of something or someone in an inferior position. For example:
"Being seen in that type of establishment is beneath me" (= I consider that establishment not to be equal to my standards).
That more abstract, usually less physical quality, when intentionally used by the speaker/author to describe a physical situation, gives the word a bit more poetic, literary feeling. Also, as with underneath, it can create a feeling of stealth, of intention, of figuratively hiding something as opposed to hiding an object; this feeling is not really connoted as much in under or below.
Consider reading in a story: "The maid had worked for the family for years, obliterating dust from every corner of their mansion. Being tidy, she kept her overcoat on a hook next to the kitchen door and her broom and dustpan in a closet under the back stairs." (= simple position; you could also say beneath the back stairs for alliteration, but it's a closet with a broom, nothing special).
Versus: "The maid knew where the family's skeletons were buried, but, in truth, how well did they know her? She maintained a tidy home and proper demeanor at all times. Little did the family know that she hid her own secrets in the closet beneath the staircase." Here beneath gives a more literary tone; it suggests something potentially physical -- like a note or diary -- but also possibly figurative, non-literal, like secret knowledge, blackmail, or maybe she quietly sobs in that closet during moments of emotional breakdown, and no one's the wiser; etc. -- it plays with the liminality between the physical and the ethereal much better than under does. Also, beneath in the latter context takes on some of the connotation of underneath, namely something hidden: hidden knowledge, hidden resentment, hidden emotions, hidden motives; however, with only two syllables, beneath feels cleaner and more figurative than the clunkier, tri-syllabic underneath, which feels much more descriptive of something physically hidden.
I would say, for example, "he hid the weapon underneath his bed," but "she hid her exasperation beneath a fake smile".
Below to me -- yes, in the above/below binary -- is also a bit stationary.
"You can find the vinegar below the sugar in aisle 3." And:
"Books on the American Revolution sit on the shelf above books on the U.S. Civil War."
Also with above-below, there seems to be an understood or implied relationship. "One shelf is above another. One shelf is below the other."
Under and underneath both feel awkward in this example. Beneath feels like too much -- like someone trying to sound fancier than they need to; I would give small side-eye to hearing "One shelf is beneath another" -- it gives a "no duh" feeling -- like they're trying to say something intelligent that is obvious, whereas above/below does not give me that feeling, it would merely be the speaker indicating where one thing is in relationship to another (e.g., where an object is), on which shelf, plain and simple. Clear and clean.
Under -- in the over/under duality -- under is imbued with a strong sense of motion or movement that words like below generally lack and beneath may or may not lack depending on the context.
"One box is below another," but "The child crawled under the table."
Most native speakers would not say "The child crawled below the table." It just sounds weird. More examples:
"The child's ball rolled under the couch," and
"The rat ran under the porch." (Most would not say "The rat ran below the porch"). You could also say "The rat ran underneath the porch," really emphasizing the feeling in the listener's or reader's mind that the rat is now hiding under the porch. In this instance you could also say "The rat ran beneath the porch." !!! Why? Because beneath can connote below in a more stationary way (1) in a situation where there's not an already-implied or -understood expected relationship between the objects and/or (2) where pairing the stationary below with a verb of movement ("ran") just feels awkward. Beneath works because of its overlapping meaning of below (lower than) and under (movement while lower than) it can be used with movement.
So movement/motion is at play with over and under. In the above versus over distinction, now we're talking about something we consider to be stationary (at least at that moment) in relative position to something else:
"A flag flew above the castle" (= more stationary) versus
"A ball flew over the fence" (was above and crossed the fence in a trajectory).
Over/under denote motion, movement, trajectory; above/below denote a more stationary position of one thing relative to another. Here's another example:
"It was such a loud morning! The police helicopter hovered above our neighbor's house while a jet flew over our neighborhood. We didn't have a moment's peace!" Above gives that feeling of stationary relative position versus over describing something in motion.
These are my thoughts at least. I will say while I'm no linguist, I do spend a lot of time translating text between English and French, and the two languages' prepositions -- how they work, how they pair with verbs, the denotations versus connotions they give -- at times could not seem more different. It can be very difficult to translate something seemingly so obvious in one language into the other and have it feel natural.
Overall, I'd say that in English we have what one might call "stronger" prepositions -- as parts of speech they have lots of independence in terms of pairing them up with different verbs in ways that convey a lot of meaning, whereas French has "weaker" prepositions that can be paired only with certain verbs and which generally carry much less of the meaning of the sentence than their verbs do. In the above examples, the sense of motion and movement versus stationary or the slight connotation of whether the speaker knows of something or if something is initially hidden or unknown carried by the prepositions does not translate easily into French, where the verbs themselves rather than prepositions would hold such meaning.
Anyway, thems my thoughts. Not sure if it helps anyone, but I thought I'd share since I'm a language lover and nerd. Here are my final examples, and please feel free to disagree with me!!!
- A river ran below the hillside town.
- A river ran beneath the hillside town.
- A river ran under the hillside town.
- A river ran underneath the hillside town.
In these examples, the image I as the writer have in my head are as follows:
- The position of the two things is stationary: the river is topographically lower relative to the higher town. Their relative positions are not changing. It reads like background description, nothing more. End of story.
- Same as above; although a slight question arises in my mind: Is the river simply lower than the town that's on the hill side? Or is the river flowing under the ground directly below/beneath the town? Beneath, to me, is a more ambiguous choice. In some contexts, it could leave a question in the reader's or listener's mind.
- There is an underground river in the earth on which the town was built and is sitting. And it's a progressive feeling: the river was still in the process of running under this town. The sense of motion in under gives it a progressive, as-yet-unfinished, still-occurring feeling in some contexts. Nothing in this sentence suggests that the townspeople didn't know there was a river beneath their feet. Under reads as neutral movement.
- Same as #3, but gives me a connotation of hidden/unknown. Perhaps the townspeople didn't realize there was an underground river beneath their feet. To me I could also imagine a sense of recency with underneath. As in, a river had just started to flow underneath the hillside town, as in an ancient underground riverbed had somehow come back to life, and perhaps that's why no one knew yet. That's just the feeling I get with underneath.