Is “You ain’t seen nothin’ yet” a common or respectable English expression?

It's a common, longstanding American slang idiom intended to convey that no matter what you've seen, what you are about to see will far top it (whether for good or for bad!). It has associations with pop-music and black American culture and expression, but it's a little dated --it has a retro feel to it these days.

Deliberately ungrammatical constructions are often used in pop music and advertising slogans because they are more memorable, convey a sense of "flavor" and seem more fresh and immediate. In particular, ain't is often used as an intensifier of the negative rather than a reversal of it (the use of double negatives to intensify rather than neutralize negativity is incorrect in "standard" English, but common in black American dialect and many other non-standard English dialects).

"You haven't seen anything yet" does have the same superficial meaning, but utterly lacks the attitude of bravado and over-the-top exaggeration. If you tell your friend who hasn't traveled "you ain't seen nothing yet," you are boasting to him about the wonders that await him. On the other hand, if you tell him "you haven't seen anything yet," you're just belittling his past experiences. The latter expression is literal, whereas the slang expression is promotional.

I was a bit puzzled with the expression, “You ain’t seen nothin’ yet.” Isn’t this a double negative, which I learnt to be an affirmative statement in high school almost 70 years ago?.

The meaning is akin to "you haven't yet seen the best, most interesting, or most exciting part".

"You ain't seen nothing yet" is a common expression -- it shows up a lot of different places. For example, James Brown's 1967 hit "Soul Man":

Got what I got, the hard way

And I'll make it better each and every day

So honey don't you fret

'Cause you ain't seen nothin' yet

Here the singer is telling his "honey" not to worry about their present condition (presumably not very good) -- because the best is yet to come.

This usage, as a reassuring or motivational statement, is very common. Here's the CEO of telecommunications company telling investors that despite poor performance, they should be patient, because exciting changes are ahead:

Sprint looks poised to continue upping the ante, with [Sprint CEO Marcelo] Claure teasing more promotions and discounts to come. "You ain't seen nothing yet," Claure said, adding that the next two to three weeks has him "extremely excited."

As for the other part of your question,

What’s wrong with saying “You’ve seen nothing yet” or “You haven’t seen anything yet”?

Grammatically speaking, "you haven't seen anything yet" is a better choice (the other alternative you suggested, "you've seen nothing yet", seems a bit condescending).

But that's a very dry, robotic way of saying it. Such is the nature of English idioms (or idioms in any language, really): they add color and connotation beyond their literal meanings and constructions.

Although grammatically incorrect, the phrase has gained a sort of acceptability through being repeated over the years. I seem to remember something along those lines from an old film ('The Jazz Singer'?) about Al Jolson.

In this context ain’t is used as part of a figure of speech and is acceptable. Ain’t is the vernacular form of am not and is considered poor grammar.