"Is there class on next Wednesday?" may sound odd (and in fact I'm having a little trouble figuring out the precise grammatical function of class here), but at least in American English, it is idiomatic and would be the most common way to ask this question.
You could also omit the word on and ask "Is there class next Wednesday?"
If you ask "Are there classes next Wednesday?", it would be interpreted as asking whether classes will be held on that day in general, not as a question about your particular class.
For example, suppose I teach at a university. Next Wednesday is not a general holiday, and most classes meet as scheduled, but I am leaving town that day and will cancel my class. If one of my students asks "Are there classes next Wednesday?" I will answer "Yes." If she asks "Is there class next Wednesday?" I will answer "No."
"Is there classes next Wednesday?" is simply a grammatical error, because classes is plural but the verb is is singular.