realfavicongenerator.net will check a whole site for correct favicons. It will also generate a full set of icons from your graphic, for all mobiles, tablets etc in a zip file that can be uploaded to the server. It tells you how to make icons work well on the site.
Each size of icon can be cusomized to best suit the size, as I did here with the badilisha.org icon.
It will also generate a browserconfig.xml and site.webmanifest.