The answer to this question about this sentence has nothing to do with whether a prepositional phrase separates the subject and verb, but with whether and when none takes a singular or plural verb. Without the prepositional phrase, the sentence can be either:
(a) None was discharged today.
(b) None were discharged today.
It's usually, but not always, a style choice and matter of perception (cf: None of our dancers are as talented as the Russians) at the end of this answer:
If none as a subject refers to something that seems multiple by nature, a speaker will use a plural verb: "None of the guests have arrived yet" feels akin to "people haven't arrived yet." However, if the speaker wants to emphasize "not one," a singular verb tends to be used: "None of the guests has arrived yet! [Dammit!]" Quite often, the underlying concept is what dictates the verb: if I speak for my colleagues and say "Heather, none of us want you to go back to graduate school and leave us" (which she just did), underlying "none of us" is the concept of "we" rather than "not me, nor her, nor him, nor her, etc."
However, not all "experts" agree:
AP STYLE BLOG: "The general rule to follow is that when the noun that follows all or none is singular, you should use a singular verb; when the noun is plural, you should use a plural verb".
GRAMMAR BOOK: "Rule: With words that indicate portions–some, all, none, percent, fraction, part, majority, remainder, and so forth –look at the noun in your of phrase (object of the preposition) to determine whether to use a singular or plural verb. If the object of the preposition is singular, use a singular verb. If the object of the preposition is plural, use a plural verb."
DICTIONARIES: "None is or none are? When none refers to a singular noun, the answer is easy – is: I needed money immediately, but none was available; None of this meat is worth keeping.
But when none refers to a plural noun, doubts can arise. This is because the fallacious idea has been implanted in the minds of English speakers that none means 'not one' and therefore, like one, must have a singular verb. It conflicts with the natural tendency of the English language, which is to make the verb agree with the noun to which the pronoun refers – if the noun is plural, then the verb is plural.
"The result is that in a plural context, both a plural verb: None of my friends were there and a singular verb: None of my friends was there can be used. In general, a plural verb sounds more natural and unaffected, and is to be preferred. It has the particular advantage that an accompanying plural pronoun avoids the sexist 'he' or 'she' and the ponderous 'he or she' in cases where the sex of the referent is unknown or unspecified: None of the directors of the company stand to lose their own money. And sometimes it would simply be ridiculous to insist on the singular – for instance, where none is used in specific comparison with a plural noun: None of our dancers are as talented as the Russians."
This agreement problem is a frequent point of contention in English usage discussions.