For anyone having issues connecting to Destiny, Bungie and Microsoft are aware of the problem. The @BungieHelp Twitter account stated they are aware of issues relating to Destiny on PlayStation 3 and PlayStation 4.
We are aware of issues preventing players from connecting to Destiny on PlayStation 3 and PlayStation 4 platforms.— Bungie Help (@BungieHelp) December 12, 2016
There was an earlier outage on Xbox 360 and Xbox One. Bungie stated that issues had been resolved on those platforms, though. If you're having issues connecting to Destiny on Xbox 360 or Xbox One, it's related to the current issues with Xbox Live.
The issues preventing players from signing into Destiny on Xbox One and Xbox 360 have been resolved.— Bungie Help (@BungieHelp) December 12, 2016
There is no ETA yet as to when the PlayStation server issues will be solved. For now keep your eye on Bungie Help's Twitter feed and we'll update this article when we find the issue has been resolved.