Yes, no, sometimes.

In it's earliest years under Joseph Smith it was a closely guarded secret. Only a few select individuals participated in Polygamy before the Mormon's moved to Utah. So for the earliest times it would definitely be only the selected "upper class."

As things progressed and Polygamy became standard nearly any member of the Church could participate in polygamy. The kicker was that the "upper class" those individuals who had power often abused that power. If a Bishop were to "receive a revelation" that a woman or girl should marry him he generally would get his way. So while not specifically limited to the "upper class." It was a privilege that the upper class enjoyed more than the average member.

The best source for this is "The Life and Confessions of the Late Mormon Bishop John D. Lee"

I'm afraid the only complete copy I could find online is from a website that seems to have a good dose of crazy. But the copy of John D Lee's confession seems accurate. Check out pages 284-286 for an account of a man castrated because a Mormon bishop wanted his fiancé.

I feel compelled to note that some parts of Mr. Lee's account have been called into question. I personally think it is mostly accurate though subject to Mr. Lee's personal biases and the limitations of his memory.