I suspect a corrupt plist
Finder>Go>Go to Folder> copy and paste: ~/Library/Preferences/com.apple.screencapture.plist
delete this plist.
You can restart an see if the problem is sorted, plist are automatically re-created to default settings
If you still have no screen shots from your terminal app (applications/Utilities/Terminal.app) copy and paste:
defaults write com.apple.screencapture type png
Try your screen capture short cuts again. If all is as expected, you are done. If you do not see your screen shots on the Desktop, again from terminal copy and paste:
defaults write com.apple.screencapture location ~/Desktop