None of the six ability scores measures an ability to stay asleep, so no.
I disagree with that. Constitution could easily be used to represent depth of sleep, and wisdom could be used to see how perceptive you are while asleep. The game doesn't back up your statement.
"The game" (meaning, the rule book) gives "Go without sleep" as an example of an attempted action for which the DM may call for a CON check. If rolling well on a CON check allows you to stay awake when you haven't had adequate sleep (which it does), I don't see the reasoning behind the same roll giving you the opposite ability to stay asleep in the presence of a disturbance.
You've never been woken up by a disturbance before and known that something happened to wake you up? I have. I have also had disturbances break into my dreams and let me know what is happening while I am asleep. An example would be a dog barking in my dream shortly before waking up to a dog barking.Also, I have a difficult time accepting that the DM is going to tell the player of a sleeping PC about the disturbance that will awaken his or her PC unless the check is successful, information the PC couldn't possibly know.
WIS and specifically your WIS (Perception) check "measures your general awareness of your surroundings". When asleep, according to "the game", you are actually unaware of your surroundings no matter what your WIS check says. Rolling it is a waste of time because it measures something that doesn't exist. And again, the metagame problem exists in which the DM is asking the player for an ability check for a task the PC itself is unaware it is undertaking. In my games, ability checks are for tasks the PCs and monsters are consciously trying to do. This avoids the following type of exchange:
The text absolutely does limit ability checks to such situations!
Here's the full text (with the relevant part in boldface): "An ability check tests a character's or monster's innate talent and training in an effort to overcome a challenge." That emphasized phrase limits when a creature's talent and training are measured by an ability check.
Only "in an effort to overcome a challenge" is the result of an ability check applicable to the outcome of the situation in question. The corresponding, also true statement is that an ability check does not test a character's or monster's innate talent and training in the absence of an effort to overcome a challenge.
So when considering whether to call for an ability check to determine if a sleeping creature notices an approaching attacker, the DM should take into account whether the sleeping creature is making an effort to do so. Considering that the last action declared by the creature's player was for the creature to go to sleep, an action that is actually the opposite of staying alert and aware of one's surroundings, the answer is that no ability check is warranted in making that determination.
"The game" (meaning, the rule book) gives "Go without sleep" as an example of an attempted action for which the DM may call for a CON check. If rolling well on a CON check allows you to stay awake when you haven't had adequate sleep (which it does), I don't see the reasoning behind the same roll giving you the opposite ability to stay asleep in the presence of a disturbance. Also, I have a difficult time accepting that the DM is going to tell the player of a sleeping PC about the disturbance that will awaken his or her PC unless the check is successful, information the PC couldn't possibly know.
If a high con represents the ability to stay awake, a low con can represent the ability to not wake up easily.
You've never been woken up by a disturbance before and known that something happened to wake you up? I have. I have also had disturbances break into my dreams and let me know what is happening while I am asleep. An example would be a dog barking in my dream shortly before waking up to a dog barking.
Again, if you were truly unaware, you could not wake up at all. You have to be aware of a sound before it can wake you up. You have to be aware of someone shaking you before it can wake you up. You may not be as aware as if you are awake, but you cannot be completely unaware.
Then the only way to wake up in the game is by DM fiat, which is how I do it. (Excepting, of course, the ways of awakening someone given in the sleep spell.)
Exactly. And sometimes DM Fiat is "Make a perceptions roll. At disadvantage"![]()
Right, but this is an ability check in form only and actually has more in common with other methods of random generation at the DM's disposal such as wandering monster checks, social reaction checks, or sleeping dragons awaken on a roll of 6 on a d6 (although in 1e they also awaken if attacked). I probably haven't yet stated clearly that I hold that view.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.