Learn English – Many a teacher and many a student has/have

I think this is a great question asked by InternetUser even though it's not one regarding a point of grammar that I find particularly interesting; however, because I really like this question, I feel that I should answer it even though I shall probably be downvoted for my usual sailing against the wind. Without further ado, here is my answer: the only correct choice is your second choice above:

"A student has posted what appear to be the images of X and Y."

Even though I like most of the other answers because native speakers often say it both ways, "appear to be" is the only grammatically-correct choice. The reason for this is that "appear" is a copula in this sense like "seem" or "be". One would not use "is" if we changed "appear to be" to "is":

"A student has posted what are images of X and Y." (correct English)

"A student has posted what is images of X and Y." (incorrect English)

"A student has posted what are or appear to be images of X and Y." (correct English)

"A student has posted what appears to be or is images of X and Y." (incorrect English)

As I've stated above, "images of X and Y" is the predicate complement of the copula "appear to be". Now, just because it's a copula doesn't make or break anything, but it is linked to "to be", which is the copula that, in English, is the equivalent of an equal sign:

"I am he." : I = he and he = I. ("He is I.")

"I appear to be the king (he)." = "The king appears to be I." ("He appears to be I.")

"Oh wait, the king (he) appears to be I and I appear to be the king (he) because I am he and he is I."

Now, the fact that this is a copula only answers your question in the example you posit above; it doesn't answer every situation wherein a pronominal "what" directly precedes a verb that isn't a copula or doesn't have copular functions. For instance, in the example,

"He has committed what amount to crimes."

it would not be third-person singular "amounts" therein because the pronoun "what" replaces a "plural noun + that" construction:

"He has committed violations that amount to crimes."

Your above example could be rewritten this way:

"A student has posted images that appear to be of X and Y."

In this instance above, one can easily see that the pronoun "what" replaces "images that", which is a "plural noun + that" construction. Now, if it were rewritten with a singular noun, i.e., "image", it would be written this way:

"A student has posted an image that appears to be of X and Y."

"A student has posted what appears to be an image of X and Y."

I hope that might make sense to you. It can be quite confusing. I agree somewhat with what everyone else has said that it can be said both ways only because "native speakers" say it both ways all the time, but just because they say it both ways all the time doesn't mean that one way they're saying it isn't more correct than the other. In your original example, I would say "appear" is more correct than "appears" and I'm euphemizing here, just to let you know. Take care and good luck in your endeavors!

"Classes" is a plural noun. The verb must also take a plural form to agree with the noun.

One class has started.

Two classes have started.