IMO B

There are two clauses in this sentence.
The subject of first clause - antique ( should match with has)
Subject of second clause - market (should match with is bringing)

. Out of America's fascination with all things antique have grown a market for bygone styles of furniture and fixtures that are bringing back the chaise lounge, the overstuffed sofa, and the claw-footed bathtub.

A) things antique have grown a market for bygone styles of furniture and fixtures that are bringing
B) things antique has grown a market for bygone styles of furniture and fixtures that is bringing
C) things that are antiques has grown a market for bygone styles of furniture and fixtures that bring
D) antique things have grown a market for bygone styles of furniture and fixtures that are bringing
E) antique things has grown a market for bygone styles of furniture and fixtures that bring

In the original sentence helping verb used is HAVE which is not aligned with the subject which is America's fascination (Singular) so HAS should be used. We have only three choices with HAS. But Option C is wordy and unnecessary uses ARE in the sentence. And option E uses BRING that point that subject is plural so that’s y E is wrong.
My answer is B .

Ans must be B.

But I think the reason in the above posts do not look nice to me.

First of all, we need to identify the subject for proper subject-verb agreement.

Q. what has been grown out of america's fascination: market. So, this sentence is inverted word construction. "have" must be "has" to agree with the subject "market".

Q. what is bringing back: market (ignore the intermediate prepositional phrase). Again "is bringing" is in agreement with "market".