How do I backup data (SMS/contacts) from a device with a broken screen? I just broke the screen of my (unrooted) Galaxy S.

How do I get back my SMS and contacts? 

PS: I tried Kies but it won't detect my phone for some reason. (Kies works fine with another Galaxy I possess but not this one)
All of the solutions above require USB Debugging to be on. If it is not and you are using Android 4.0 then you can press power and volume down buttons simultaneously to obtain screenshot. If USB is not connected it will be placed somewhere on the SD card (e.g. \Pictures\Screenshots). Then you can connect USB to see the screenshot, then disconnect, navigate further and so on till you navigate yourself to enabling USB Debugging.

When USB Debugging enabled it is all easy as pie.


Provided you have SDK installed you can use ddms.bat from tools folder to get screenshots fast and convenient, adjusting any options you need and even installing apps.
Classic way - adb
I've found Moborobo to be very convenient way to make full backup


I understand this is very late and hardly helps OP, but I hope it helps someone else lurking here for the way to save data, like me.