When trying to add or update a Simple URLs entry, my publish/update button had no effect – the activity indicator next to the button does not appear and the entry is not published or updated.
I suspected a plugin conflict so from a process of elimination I have identified that this occurs when Jetpack is active, but after deactivating Jetpack the problem disappears and I can save Simple URLs entries.
I’ve enabled both Jetpack and Simple URLs on one of my sites and the publish/update functionality of Simple URLs is working. Could you double check and make sure you do not have any other plugins besides Jetpack and Simple URLs enabled? Also, check for a mu-plugins folder in your wp-content directory. If there is anything there, please remove it temporarily for testing.
ETA: Another test to do is switch to the default theme and try using Simple URLs.
Thanks Tim, It has now been identified as a bug in the After the Deadline component of Jetpack mdawaffe has managed to reproduce the error now and says there will be a fix for this in the next version of Jetpack.