Stoning an adulterer is the universally accepted legal punishment among the scholars of sunni islam (see e.g. islamqa), and I think among shia scholars as well. The material law (i.e. what is the punishment and what constitutes adultery) is quite clear; penetrative sex while married with someone you are not married to constitutes adultery, and the punishment is being stoned to death.
The procedural law, i.e. questions of burden of proof etc., is a different matter. To prove adultery via eye-witnesses would require four adult male muslim witnesses of good standing who actually saw the penis enter the vagina. I have also seen the opinion that the husband who finds his wife having sex with another man could swear four times (this is called "li'aan"; search "should make him swear four times by Allaah" on that page) to replace the four witnesses in order to match the burden of proof; the so-accused wife can herself swear four times to avert stoning. Proving her husband has been adulterous by means of li'aan is not accepted from the wife according to that link. I do not know how widespread li'aan is, but the source for it, 24:6-24:9, suggests that it is probably widely accepted. Another way of meeting the burden of proof is a confession by the offender.
To gauge the popular support for this punishment in the general population, the pew research center polled the populations of majority-muslim countries; with Turkey at 16% of the muslim population in support of stoning adulterers to death forming the least committed group, it seems like a very significant proportion of muslims worldwide is willing to support this punishment, at least by voicing that opinion in a poll. The question whether they would work towards implementing it, or tolerate it once it is enacted into law, is not really answerable from that data, of course.