When I started my laptop up the other day, nothing was functioning. I couldn't open any applications and the thing was absolutely crawling like there was no free memory. After a few restarts it started running back at speed. However, when I use it now there will randomly be icons missing from my desktop.

Things such as my computer are simply not visible on the desktop. I can still access programs via the Start menu but obviously this is annoying and perhaps the indication of a bigger problem. Can someone please tell me how to fix this?

Here are a few basic fixes:

Right-click on your Desktop and select:
Properties > Desktop > Customize Desktop > Restore Default


right click on the desktop, chose properties, go to the
desktop tab, click Customize Desktop... there you can
choose which icons you want to be shown on the desktop.



Start
My Computer
Right click, check "Show on desktop"

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As to your overall condition I would suggest:

please follow owen's Instructions and then post your log in the Spyware, Adware, Viruses and HijackThis Logs section (Not Here).