Mozilla and Opera are standard-compiliant.
Wingdings is not standard (what a surprise coming from Microsoft) because not mapped to Unicode, and so should never be used on a website.
However, do not despair, most symbols are available in Unicode : check http://www.alanwood.net/demos/wingdings.html
Important addendum (october 2014) : As of Unicode v7, released on June 16, 2014, all characters from Webdings and Wingdings have been added to Unicode. See the comment below.