Xbox One users have taken to the Bungie Forums to report issues connecting and staying connected to Destiny's servers. Bungie has responded to those affected, saying that they are both aware of the situation and are investigating the issue.
We are aware of players on Xbox One encountering issues connecting to Destiny and are currently investigating. Please standby.
— Bungie Help (@BungieHelp) September 17, 2015
It appears as though the issue is affecting users that are both hard-wired to the internet and those that are using Wi-Fi. At the moment it is unclear what is causing the issue or if it affecting both Xbox One and Xbox 360 users, most reports are currently citing the issue as being Xbox One specific.
Microsoft's Xbox Live server status is reported to be up and running. (Update: XBL core services are down as of 4:23 PM EST).