The singular possessive is used a lot more often than the plural in this phrase.

But the singular form "doctor's" is dubious without an article.

I disagree. It is a kind of bill, an area of expenditure. If you thought it was dubious, say so to begin with. Your questions are crappy.

I am talking about the number of "doctor." Would you say, for example, "Student's safety is very important?

It's not the same thing at all. I would say student safety. No possessive. And we also do this with doctor bills.

Or "students' safety," right? How about "Book's covers are important"?

Yes, "doctor bills" without the apostrophe is definitely correct.

People hearing this phrase do not glean from it the number of doctors involved, no matter how you choose to make the possessive.

I am wondering whether there is a general rule allowing the use of singular, article-less, countable nouns.

I'm afraid you will have to remain in that state of wonder.

Doesn't that suggest there is a rule against using an article-less singular countable nouns?

The reason why I suspect "doctor's bills" is ungrammatical is the same as the reason why "Book is interestinig" or "Book's cover is interesting" is ungrammatical.

The situations are not at all analogous. In the possessive doctor's bills the word doctor's functions as a specifier, which is precisely what is lacking in "Book is interesting", a specifier.

"Book's cover is interesting" is ungrammatical. What do you say about that?

I agree that Book's cover is interesting is ungrammatical. I spent all my money on book's covers is not grammatical. But I spent all my money on doctor's bills is grammatical. Why? You'll have to ask doctor.

Is it possible that "doctor's bills" is a mis-transcription of "doctors' bills"?

Do you consider the act of writing your thoughts on the page an act of transcription?

I was saying that maybe some people misplace the apostrophe in writing.

"some"? My very first comment says that the singular possessive is used a lot more often than the plural. So you're basically saying that what the vast majority of people do here is wrong.

The fact is, I asked the question elsewhere and many of the speakers there don't like "doctor's bills."

One might question your sample size.

In comments you seem to have already answered your question. You should update it with what you've concluded—and clarify what your actual question is at this point. (For one, you've made it clear that you're not talking about a possessive but an attributive noun. But that's not the assumption anybody would make from the existing text of the question.)

merriam-webster.com/dictionary/doctors'%20bills