"One of the most crucial problems that..."—"exists" or "exist"?

Short answer

Either sentence is fine. The notional Subject of the verb exist is the plural noun phrase the most crucial problems. However, in English the word one which occurs earlier in the larger noun phrase can optionally override the expected subject-verb agreement in the relative clause causing a singular verb form.

Full answer

1. One of the most crucial problems that exist in industrial countries such as ...
2. One of the most crucial problems that exists in industrial countries such as ...

Based solely on the grammar there are two possible readings of these sentences:

• A: There is a group of crucial problems which exist in industrial countries. This problem is a member of that group

• B: This problem is a member of a group of crucial problems. This problem exists in industrial countries.

Common sense tells us that the writer wants to convey meaning A, and not meaning B. We could model the sentences like this according to their meaning:

• A: One of [the most crucial problems that exist_ in industrial countries] is
• B: One of [the most crucial problems] that exist_ in industrial countries is

Here we can see that with meaning A, the notional Subject of the verb exist is the plural noun phrase the most crucial problems. In B, the notional Subject of the verb exist is the singular noun phrase one of the most crucial problems.

Now it would be nice, wouldn't it, if sentence (1) always meant A, and sentence (2) always meant B. However, that's not the case. The fact is that native speakers, both in speech and writing, often use sentences like (2) to express meanings like A. The reason is that the word one often overrides normal verb agreement. In other words although sentence (1) definitely expresses A, sentence (2) would be used to express either A or B.

Here is what a vetted grammar source, based on real data, the Cambridge Grammar of the English Language has to say about this:

. . . The relativized element in these examples is object. Where it is the subject that is relativized, the expectation would be that the number of the verb would be determined by the antecedent, giving a plural verb in Type I, and a singular in Type II. In practice, however, singular verbs are often found as alternants of plurals in Type I:

[22]

• i. He's [one of those people who always want to have the last word]. -- (Type I )

• ii. He's [one of those people who always wants to have the last word]. -- (Type I )

• iii. He's [one of her colleagues who is always ready to criticize her]. -- (Type II )

Examples [i] and [iii] follow the ordinary rules, but [ii] involves a singular override. It can presumably be attributed to the salience within the whole structure of one and to the influence of the Type II structure (it is in effect a blend between Types I and II ). But it cannot be regarded as a semantically motivated override: semantically the relative clause modifies people. This singular override is most common when the relative clause follows those or those + noun.

In this kind of situation, thinking about the grammar is not the most useful guide. Even working out what the grammar is will take you quite some time. Go with your intuition!

^{Grammar note:}

^{You may well be wondering why I used the term notional Subject instead of just Subject. The reason is that both of these noun phrases occur outside of the exist clause. They are potential antecedents for the Subject of the relative clause, but they do not occur within the relative clause itself.}

^References:

^{The Cambridge Grammar of the English Language, Huddleston & Pullum [et al], 2002.}