I'll just quote myself from early last year to show the single player problems I ran into. I did a Heroic runthrough of the series and then replayed it on Legendary twice. I've never not run into at least one game ruining door bug on any of my playthroughs of 2 and 3 on this collection.
Played through all the games in the collection on Legendary over the Christmas holidays. Still had game ruining bugs in Halo 2 and 3 where sometimes doors wouldn't open to let you continue and you had to start the whole level over. Here's the ones I personally encountered.
After the Scarab segment in the final level of Halo 2 "The Great Journey", you clear a room of Brutes before moving on to the final area. A second wave is supposed to come out of the door that's the way forward after you clear the first wave. Sometimes they don't, and you're screwed. Happened the first time I played it at release too.
Final level of Halo 3, where you scale the pyramid with all the flood forms. Spark's supposed to open the door after you clear them all and some dialogue from Johnson triggers, but sometimes that dialogue doesn't trigger, Spark doesn't open the door, and you're screwed. This one also happened to me at release. At least this bug is near the beginning of the level.
In Halo 2's "Gravemind", there's a room with 2 Hunters towards the end of this long level. After killing them, Cortana is supposed to pop up and talk, triggering the door opening and more enemies appearing. Sometimes she doesn't, and you're screwed. Thankfully, this one didn't happen on my recent playthrough on Legendary, but that's probably just luck. I doubt they fixed it.
As it stands, I haven't played through either of these games without it happening at least once. Never had problems with the originals on the 360. Thankfully the other games in the collection don't seem to be plagued with it. It's such a shame considering how great of a collection it is.
I also encountered no less than half a dozen crashes when examining terminals, though I think those were fixed pretty quickly. I never even touched multiplayer and I can say with confidence that the single player was definitely NOT OK at launch.