Following a free upgrade from Win7 to Win10 a few days ago I found that I'd lost my IE icon and despite me saying NO to Edge being my default browser, Microsoft ignored me and Edge was now my default browser.
Here are the solutions:
(1) to get your icon back go here: https://support.microsoft.com/en-us/kb/945402] See PS.
[This is for Win7 - Win8 users must follow the link in the very first sentence.]
I used the third method - "alternative method to method 1". No problems at all, even for me.
[If you lose it again in Win10 go to here: http://windows.microsoft.com/en-gb/internet-explorer/pin-internet-explorer-start-taskbar#ie=ie-11
(2) to make IE your default browser, click on Start (the new icon) and then Settings > System > Default Apps. Again, this is very simple.
PS Presumably poor old Microsoft were referring to previous legal decisions in the EU and elsewhere when they talk about compliance reasons here - it wasn't then that they are simply trying to force users to give up IE in favour of Edge - is it?
"Important Windows 7 no longer supports the ability to modify the registry to get the special Internet Explorer icon to appear on your desktop. This change was made to ensure Internet Explorer could be removed easily for compliance reasons."