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Goodevening,
I have had my Sony Xperia Z for 6 days and it's been great up till this evening.
When I touch to open applications, it doesn't respond and this happens with all apps. However the on-screen buttons and swiping through the app drawer/home screen were still working without problem and if after many tries an app opens, the in-app operation was smooth.So the problem is just in the opening apps from home screens and app drawer.
After a while I've come to notice that the apps only opens when I touch the upper left corner of the icon of the apllication. This is a bit frustrating.
Any help/suggestions would be appreciated.
restart your phone and try again
also do you have any taskillers or battery apps? if you do, uninstall them > restart your phone and check again
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if the above sugestion doesn't work then repair your phone with SUS
Update Service (SUS)
http://www.sonymobile.com/gb/tools/update-service/
http://www-support-downloads.sonymobile.com/Software%20Downloads/Update_Service_Setup-2.11.12.5.exe
for future reference
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355
Don't forget to mark the Correct Answers & Helpful Answers
I had only DX battery saver, no task killer app, I've uninstalled the Dx battery saver and resarted but it's still the same problem
repair your phone with SUS (link above) and check again
Don't forget to mark the Correct Answers & Helpful Answers
Thanks, this helped solve the problem
Anytime 
Using the update service did help for a while but this afternoon, the same problem started again
then it is an app acting up, try to find out which app is it, most likely a
taskiller, antivirus, task manager, ram apps, battery apps, booster apps.
Update Service (SUS)
http://www.sonymobile.com/gb/tools/update-service/
http://www-support-downloads.sonymobile.com/Software%20Downloads/Update_Service_Setup-2.11.12.5.exe
for future reference
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355
Don't forget to mark the Correct Answers & Helpful Answers