Just a quick compilation of the bugs me and my friends have experienced over the first few days. This is ordered in terms of general severity. This is all on the Xbox One version, not sure if these are present on PC or Next-Gen consoles.

1. If you have the Raygun or any other pistol and go down with it in your hands, you may lose the gun and have your secondary changed to Fists. You cannot get another secondary to replace your fists as it will always swap your other gun. You will continue to have the pistol when you are downed.
2. Accepting a Sentry Turret legendary reward from the trials crashes your game (at least in co-op.)
3. If you fill the vial with the liquid nitrogen, pick it up, and then go to the dark aether before using it, you are unable to get the ice upgrade for the D.I.E. as the chest no longer allows you to pour the vial. You cannot continue the Easter Egg.
4. If you open the box containing the fuse and wait too long to get it (unclear how long exactly) it will despawn and you cannot continue the Easter Egg.
5. If both Megaton launchers are sucked into the containment chamber at the same time, the audio telling you to enter the computer room does not play, although the Easter Egg will progress when you all enter.

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This is all I remember for now, feel free to comment with what you've experienced so far and I'll add it to the list.