If you right click on the icon on your desktop, and choose Properties, on the General tab there's a choice to "Change icon."
You can browse for one of the several icons that are provided by Windows, or you can pick one from those that "came with the program," although finding the available ones among the program-specific ones can be "iffy." The icons that come with programs that you install often are "imbedded" in the program and sometimes don't show up easily.
If you see the icon sometimes and it goes away at other times, it probably is linked to your desktop via a cookie. If you see it, it's probably coming from browser temp files somewhere. If you clear temp files or cookies, the icon could disappear and you'd get a default icon - probably the one picked by your browser.
To have the icon "permanent" you likely would need to find the icon and save it as a separate file somewhere on your own computer, and then set it as noted above.
You can look in your
C:\Documents and Settings\username\Local Settings\Temporary Internet Files
and click at the "Name" on the top bar to sort by Name and find the "favicon[1].ico" from mudcat and copy it to some permanent place on your computer, and you should be able to "attach" it to your shortcut using the "Change Icon" menu.
Note that Windows may warn you that "there are strange and dangerous things there" in the Temp folder, and you shouldn't go there; but if you haven't been cruising profligately you should be safe enough getting in and saving the icon.