I loaded the 3.x quickstart to check out the template features. There was no search icon, and since search was important to me I went looking.
Seems that there is no search position available in the position selection. It was in the code but missing from the templateDetails.xml file in the template. So I added this line under positions <position>search</position>. All is well now and search all works fine with the standard Joomla search module published in this position.
Might help others, but I think that line is missing from the quickstart package.