For all with the issue "offline" mode doesn't work for them with games:
if the console is not set as home console, "offline" doesn't work
setting the console to home console while the outage is ongoing doesn't work either because the setting needs to be checked by the servers (security check - only one home console allowed)
games that are multiplayer or have some sort of always online implementation (Destiny, Powerstar Golf) don't work because the service is down that support those features.
Strictly offline single player games (that need no online resources at all) work fine in "offline mode": I personally tried and can confirm those games are working: BF4 SP campaign, Halo5 SP Campaign, GTA V SP, DA:I, Dishonored DE, ROTTR, Both Wolfenstein games
(I am all digital!)
Edit:
Just tried to start a bunch of other games: Borderlands 2 and Pre-Sequel work (you will be asked if you want to play offline when the game starts), Fallout 4 works too, Ori And The Blind Forest works, Witcher 3 works
Edit2:
Apps: Netflix doesn't work with offline, Amazon Instant Video doesn't work, BluRay does work!