The right way to look into this sentence would be to remove the preposition phrases so that the subject and verb can be clearly understood.
Lets strike out the propositional phrases and see if that'll help:-
Out of America's fascinationwith all things antique has/have grown a marketfor bygone styles of furniture and fixtures that is/are bringing back the chaise lounge, the overstuffed sofa, and claw-footed bathtub.
As you can see, its now easier to figure out the right answer.
fascination->singular->has
market->singular->is
...
Lets strike out the propositional phrases and see if that'll help:-
Out of America's fascination
As you can see, its now easier to figure out the right answer.
fascination->singular->has
market->singular->is
...
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