Lollipop 5.1, completely safe from stagefright? I am currently updating my Android phone (a Moto G) to Lollipop 5.1.  I previously had taken action to prevent Stagefright from happening to my phone (setting multimedia messages to only manual downloads).  The update said that Lollipop 5.1 included a fix for Stagefright.  I am wondering what this fix is and if my phone is now completely safe from Stagefright (i.e., I can set multimedia messages to automatic download again).
You can use Zimperium's Stagefright Detector app to test your device for the known attack vectors. The bugs were not patched upstream until Android 5.1.1, but it is possible (particularly based on the change notes) that Motorola backported security fixes to their 5.1 release, so the version of Android you have is not necessarily the best indicator.

NB: One of Zimperium's researchers is Joshua Drake, who disclosed the Stagefright vulnerabilities to Google and submitted patches to fix them. It is therefore reasonable to conclude that the tests in their app are more than likely accurate.