I got the notification that all my Joomla installs were ready for the 3.7 update. I started with test sites and then moved my way to my productions sites.
I have updated 12 sites so far and no issues. I even I have felt that the sites were running quicker.
I truely feel the the 3.X versions of Joomla started out slow and have gotten faster.
I feel like it is safe to update your Joomla installs.
I hope this helps someone else out there…