Does "a [x] of [y]" take a singular or plural verb?

What kind of verb, singular or plural, goes with phrases like "a record of [singular or plural noun]", "the use of", "the time of"? Does the choice depend on the following nouns (a record of nouns) or on the 'first' nouns (record, use, time)? Or do these phrases obey the rule of proximity (as does "a variety of")?

I have found different example sentences on the internet (source: newspapers or magazines) with different uses. What is correct?

Examples:

• there were a record of other issues;
• there were a record of 28,000 runners;
• there was a record of these statements.

First, you're confusing the issue a bit with a misuse of "a record" in the adjectival sense. "There was a record of 28,000 runners participating" technically¹ means that documentation existed for this number of runners – perhaps there were more runners, but the remainder were not documented. Here, the subject is "record", so the verb is singular. If you're trying to unambiguously remark on the fact that 28,000 is a large number, you need to leave out the "of": "There were a record 28,000 runners participating." Here, the subject is "runners", so the verb is plural.

^{1 In practice, it's an ambiguous statement, because lots of people use "a record of" incorrectly.}

In the general case of "a [x] of [y]", the verb should agree with [x], unless it's one of the special cases (e.g. a lot of) where [x] itself is sorta-kinda plural in meaning, and so can take a plural verb. To figure it out, you can remove the extra words and see if the result sounds correct.

• There was a record of these statements. (There was a record.)
• The use of apples instead of pears was unexpected. (The use was unexpected.)
• It was a time of mass uprisings and protests. (It was a time.)
• There were a lot of people present. (There were a lot.)
• A variety of options were available. (A variety were available.)

It looks as if you're asking "Does the phrase 'a record of several things' take a singular or plural verb?" If so, the answer is "Singular"; the verb agrees with the noun, not with the description. Proximity has nothing to do with it. (This should also be closed as general reference).

If you're asking something else, please clarify.

These are genitive constructs, where the "of x" is a restriction applied to the noun. The purpose of the genitive is to restrict the noun, it is not the object of the verb itself. (Strictly since you are using a copula here, I should say it is not the complement of the noun, but you get the idea.)

There was a record of lots of people.

The complement of the verb is "record." You can see this by dropping part of the sentence.

There was a record.

This is clearly correct. The rest of the sentence "of lots of people" simply restricts which type of record you are talking about.

See Can "what kind" be plural? for a similar discussion.

Usually, in sentences like this, the verb must agree with the first noun. There are a few nouns ("a lot of", "a kind of", "a herd of") where the verb will always or often agree with the second noun.

Note that "a record of" can have two meanings: as in "documentation" or as in "world record". For the first meaning ("a record of other issues"), the verb should always agree with "record". For the second meaning ("a record of 28,000 runners"), my feeling is that both choices can be grammatical.

Whether the verb should agree with the first or second noun should depend on which noun is associated with the action of the verb (often, either choice is acceptable).

A record of thirty runners were treated for heat stroke this year.

A record of thirty Atlantic hurricanes was set that year.